JC-NRLF 


MACHINE  SHOP 
MATHEMATICS 


m 


THE  LIST 


•• 


IN  MEMOR1AM 
FLOR1AN  CAJORI 


*-/  . 


MACHINE-SHOP 
MATHEMATICS 


BY 

GEOKGE  WENT  WORTH 

•  t 

DAVID  EUGENE  SMITH 

AND 

HERBERT  DRUERY  HARPER 


GINN  AND  COMPANY 

BOSTON    •    NEW  YORK    •    CHICAGO    •    LONDON 
ATLANTA    •    DALLAS    •    COLUMBUS    •    SAN    FRANCISCO 


COPYRIGHT,  1922,  BY  GINN  AND  COMPANY 

ENTERED  AT  STATIONERS'  HALL 

ALL  RIGHTS  RESERVED 

422.4 


Tgfte   gtfrenatum 

GINN  AND  COMPANY  •  PRO- 
PRIETORS •  BOSTON  •  U.S.A. 


PEEFACE 

Purpose  of  the  Work.  This  work  has  been  prepared  to  meet 
the  needs  of  students  who  expect  to  become  machinists,  either 
in  the  special  line  of  automobile  construction  or  in  the  more 
general  lines  of  the  machine  shop.  It  is  therefore  strictly  limited 
in  scope  to  the  needs  of  those  who  are  entering  upon  this  kind  of 
work,  and  it  treats  only  of  such  topics  as  experience  has  shown 
are  demanded  by  the  practical  machinist  who  is  determined  to 
advance  in  his  vocation. 

Work  Presupposed.  The  student  is  supposed  to  have  covered 
the  work  laid  down  in  the  authors'  "  Fundamentals  of  Practical 
Mathematics,"  or  its  equivalent,  and  therefore  to  be  familiar 
with  the  use  of  whole  numbers,  common  fractions,  decimals,  per 
cents,  proportion,  and  the  common  tables  of  measure  as  applied 
to  practical  problems.  While  it  is  not  an  absolute  essential 
that  the  student  should  have  mastered  the  slide  rule  or  should 
be  thoroughly  acquainted  with  the  elements  of  trigonometry 
and  with  the  metric  system,  it  is  desirable  that  he  should  have 
at  least  a  fair  working  knowledge  of  these  subjects. 

Topics  Considered.  A  glance  at  the  Contents  will  show  the 
topics  considered,  the  relative  amount  of  attention  given  to 
each,  and  the  sequence  in  which  they  are  taken  up.  In  general 
it  may  be  said  that  the  choice  of  topics  and  the  time  allotted  to 
each  are  conditioned  by  the  actual  needs  of  the  student,  while 
the  sequence  is  based  chiefly  upon  the  question  of  relative  diffi- 
culty, although  due  attention  has  been  given  to  the  dependence 
of  one  topic  upon  another.  The  first  thing  that  is  needed  is  a 
knowledge  of  the  measuring  instruments  actually  used  in  the 

iii 


iv  PREFACE 

machine  shop,  since  without  this  knowledge  the  rest  of  the  work 
is  meaningless ;  the  second  topic  relates  to  speeds  and  feeds,  this 
being  the  first  thing  that  the  student  meets  in  the  use  of  a 
machine ;  the  third  topic,  tapers  and  taper  turning,  follows  natu- 
rally, and  so  on  throughout  the  book. 

The  authors  believe  that  they  have  succeeded  here,  as  in  their 
earlier  book  in  this  field,  in  eliminating  nonessentials,  in  empha- 
sizing the  great  principles,  and  in  presenting  the  matter  in  a  new 
but  perfectly  natural  form,  with  definite  and  valuable  applica- 
tions which  initiate  the  student  into  the  actual  work  of  the  shop. 
They  hope  that  their  efforts  will  meet  with  the  approval  of  teachers 
and  students  alike. 

Acknowledgment.  The  authors  wish  to  express  their  thanks 
to  the  following  manufacturers  who  have  given  permission  to  use 
the  illustrations  shown  on  the  pages  mentioned:  South  Bend 
Lathe  Works,  South  Bend,  Ind.,  p.  8 ;  K.  K.  LeBlond  Machine 
Tool  Co.,  Cincinnati,  Ohio,  p.  62 ;  Cincinnati  Milling  Machine  Co., 
Cincinnati,  Ohio,  p.  63 ;  Brown  and  Sharpe  Manufacturing  Co., 
Providence,  K.  L,  pp.  70,  81,  91,  99,  105,  and  115 ;  Meisel  Press 
Manufacturing  Co.,  Boston,  Mass.,  pp.  97,  103,  and  112 ;  Niles- 
Bement-Pond  Co.,  New  York,  N.Y.,  p.  110;  Hoeffer  Manufactur- 
ing Co.,  Freeport,  111.,  p.  118 ;  Ohio  Machine  Tool  Co.,  Kenton, 
Ohio,  p.  124 ;  Lynd-Farquhar  Co.,  Boston,  Mass.,  p.  126 ;  Garvin 
Machine  Co.,  New  York,  N.Y.,  p.  128;  and  the  Putnam  Machine 
Works,  Fitchburg,  Mass.,  p.  132.  Other  well-known  manufacturers 
supply  similar  machines  of  a  high  degree  of  precision,  and  they 
would,  no  doubt,  have  been  equally  willing  to  give  permission  to 
use  their  illustrations  had  it  been  requested. 


CONTENTS 


CHAPTER  PAGE 

I.   MEASURING  INSTRUMENTS 1 

II.   SPEEDS  AND  FEEDS .     .     .     .  9 

III.  TAPERS  AND  TAPER  TURNING 21 

IV.  SCREW  THREADS 35 

V.   INDEXING  AND  SPIRAL  CUTTING 61 

VI.   GEARS      . 85 

VII.   REVIEW  PROBLEMS 117 

TABLES  FOR  REFERENCE 137 

INDEX  159 


MACHINE-SHOP  MATHEMATICS 

CHAPTER  I 

MEASURING  INSTRUMENTS 

Measuring  Lengths.  In  measuring  short  distances  we  take 
each  distance  with  a  pair  of  dividers,  transfer  it  to  a  steel 
ruler,  and  then  read  off  the  length.  Since,  however,  rulers 
are  seldom  graduated  beyond  g1^",  this  method  is  not  accu- 
rate enough  for  fine  work. 

For  finer  work  a  diagonal  scale,  which  applies  the  principle 
of  parallel  lines  to  measuring  lengths,  is  sometimes  used  by 
draftsmen.  In  the  scale 
here  shown  the  distance 
from  the  vertical  line  1 
to  0  is  1".  From  the  ver- 
tical line  1  to  the  point 
where  diagonal  line  0 
cuts  horizontal  line  8, 
the  distance  is  1.08" ;  8  6  4  2  °  1 

/  PART    OF    A    DIAGONAL    SCALE 

to  the  point  where  di- 
agonal line  5  cuts  horizontal  line  6,  the  distance  is  1.56";  and 
so  on,  so  that  by  this  method  we  can  readily  measure  to  0.01". 
For  the  work  required  of  a  skilled  machinist,  however,  a 
higher  degree  of  accuracy  than  is  obtainable  by  either  of  the 
two  methods  given  above  is  necessary.  A  method  of  obtain- 
ing this  greater  precision  by  the  application  of  the  principle 
of  the  screw  thread  is  explained  on  page  2. 

1 


MEASURING  INSTRUMENTS 


Caliper.  A  caliper  is  an  instrument  for  measuring  thick- 
ness or  diameters.  One  of  the  most  common  forms  is  a  pair 
of  compasses  with  legs  curved  inward  for  measuring  outside 
diameters  and  the  thickness  of  plates.  The  legs  may  also 
curve  outward  for  measuring  inside  diameters. 

Micrometer  Caliper.  A  caliper  of  the  type  shown  in  this 
figure  is  called  a  micrometer  caliper,  or  simply  a  micrometer. 


Reading : 

0.300"  (numbered  graduation  on  E) 
.025"  (unnumbered  graduation  on  E) 
.003"  (thimble  reading  on  F) 

0.328"  (reading  of  setting) 


MICROMETER    CALIPER 

A,  anvil ;  B,  screw ;  (7,  frame ;  D,  lock  nut ;  E,  barrel ;  F,  thimble ; 
G,  ratchet  stop 

In  this  instrument  the  principle,  which  is  also  used  in 
other  types  of  measuring  instruments,  is  that  of  a  screw  with 
40  threads  to  the  inch.  At  each  revolution,  therefore,  the 
screw  moves  lengthwise  -fa",  or  0.025".  If  by  means  of 
marks  on  the  instrument  we  can  tell  when  the  screw  has 
made  ^  of  a  revolution,  we  shall  know  when  it  has  moved 
lengthwise  ^  of  -fa",  or  0.001". 

In  the  caliper  shown  above  the  thimble  is  attached  to  the 
screw  and  moves  along  the  barrel.  When  the  screw  is  against 
the  anvil,  the  0  on  the  thimble  coincides  with  the  0  on  the 
barrel.  The  barrel  is  graduated  to  -fa",  the  numbered  lines 
being  0.1"  apart.  Each  revolution  of  the  screw  opens  or 
closes  the  caliper  ?y,  or  0.025",  and  therefore  moves  the 
thimble  one  graduation  on  the  barrel.  Each  of  the  25  gradu- 
ations on  the  thimble  indicates  of  0.025",  or  0.001".- 


MICROMETER  CALIPER 


3 


Directions  for  Reading  a  Micrometer.  In  reading  a  microm- 
eter caliper  of  the  type  shown  on  page  2  proceed  as  follows  : 

Count  each  numbered  graduation  on  the  barrel  as  0.100", 
each  unnumbered  graduation  as  0.026",  and  each  graduation  on 
the  thimble  as  0.001".  The  total  is  the  reading  of  the  setting. 


MICROMETER    READINGS 

The  reading  of  the  figure  at  the  left  is  0.125";  that  of  the  one  at  the 
right  is  0.188" 

The  student  should  carefully  examine  the  above  illustration 
and  the  one  given  on  page  2  and  should  verify  the  readings. 
It  is  only  by  actually  using  the  instrument,  however,  that  he 
will  acquire  the  necessary  facility  in  reading  it. 

Exercises.    Reading  the  Micrometer  Caliper 

1.  If  the   thimble   of  a  micrometer  caliper  is  given  five 
revolutions,  how  far  has  the  caliper  opened  or  closed  ? 

2.  What  is  the  diameter  of  a  piece  of  stock  when  the  read- 
ing of  the  numbered  graduations  on  the  barrel  is  7  and  the 
thimble  reading  is  22  ? 

3.  Reduce  ^'f  to  a  decimal  to  the  nearest  0.001"  and  tell 
how  you  would  set  the  caliper  to  have  this  reading. 

4.  How  many  turns  must  you  give  the  thimble  to  measure 
a  thickness  of  4"  ? 

o 

5.  If,  when  the  screw  is  against  the  anvil,  the  thimble  is 
given  |  of  a  turn,  what  is  the  reading  of  the  micrometer? 

6.  How  many  turns  must  you  give  the  thimble  to  have  a 
reading  of  0.375"?  of  0.261"?  of  £"?  of  0.955"? 


4  MEASURING  INSTRUMENTS 

Vernier  Caliper.  Calipers  are  also  made  with  a  device  known 
as  a  vernier,  by  the  aid  of  which  we  can  read  measurements  to 
0.001".  Such  a  caliper  is  illustrated  in  the  figure  below. 


VERNIER    CALIPER 

A,  beam ;  B,  sliding  head ;  C,  clamp ;  D,  solid  jaw ;  E,  sliding  jaw ;  F,  vernier 

The  vernier  is  based  upon  the  principle  that  although  the 
eye  cannot  easily  decide  upon  a  fractional  part  of  a  small 
space,  it  can  readily  tell  when  two  lines  appear  to  coincide. 

On  beam  A  each  unnumbered  division  is  -fa",  or  0.025", 
while  on  the  vernier  F  each  unnumbered  division  is  0.024". 
Hence  25  divisions  on  F  are  together  equal  to  24  divisions 
on  A,  and  each  division  on  F  is  0.001"  shorter  than  each 
division  on  A.  If,  therefore,  0  on  F  is  slightly  to  the  right 
of  a  mark  on  A,  but  1  on  F  coincides  with  a  mark  on  A,  then 
we  see  that  0  is  0.001"  to  the  right  of  the  mark  at  its  left. 

In  the  above  illustration,  line  2  (indicated  by  the  arrow) 
on  F  coincides  with  a  hue  on  A.  The  reading  of  the  scale  on 
A  is  1.000"  +  0.500"  4-  0.025",  or  1.525".  Adding  the  0.002" 
from  F,  we  have  1.527"  as  the  reading  of  this  setting. 

For  the  clear  understanding  of  this  work  the  student  should  have  a 
vernier  caliper  in  his  hands,  setting  it  for  various  widths  of  objects  to 
be  measured  and  reading  off  the  measurements. 


VERNIER  CALIPER 


12345 


Reading  the  Vernier.    The  detail  of  part  of  the  scale  of  a 
vernier  caliper  is  shown  below  and  illustrates  more  clearly  than 
the  figure  on  page  4  the  method 
of  reading   the  vernier  caliper. 
Since  the  smaller  of  the  numbered 
divisions  on  A  represent  0.1",  we 
first  see  that  we  have  0.8"  plus 
two    unnumbered    divisions    of 

„    ~~~,,  T  ^   r^r-^i,    .  PART   OF    THE    SCALE    OF    A 

0.020"  each,  or  0.850"  in  all.  VERNIER  CALIPER 

We  also  see  that  line  14  (in-  A>  scale  on  beam;  F%  vernier  scale 
dicated  by  the  arrow)  on  the 

vernier  scale  coincides  with  a  line  on  the  scale  on  the  beam. 
We  therefore  add  0.014"  to  0.850",  the  result  being  0.864", 
the  reading  of  this  setting. 

Micrometer  Caliper  with  a  Vernier.  By  combining  the 
vernier  and  the  micrometer  we  can  obtain  readings  to  0.0001". 
In  the  figure  here  shown  a  ver- 
nier of  ten  divisions  is  marked 
on  the  barrel  A  and  occupies 
the  same  space  as  nine  divisions 
on  the  thimble  B.  The  microm- 
eter  reading  is  0.137",  accord- 
ing  to  the  explanation  given 
on  page  3. 

Since  line  5  (indicated  by  the  straight  arrow  at  the  left 
on  the  barrel  A)  of  the  vernier  coincides  with  a  line  on  the 
thimble,  we  add  0.0005"  to  0.137"  and  the  reading  of  this 
setting  is  0.1375". 

Thus,  by  the  aid  of  a  relatively  simple  device  we  can  find 
the  diameter  of  a  rod  or  the  thickness  of  a  sheet  of  metal 
to  a  higher  degree  of  accuracy  than  would  seem  possible 
if  one  did  not  know  the  ingenious  mathematical  principle 
involved  in  the  vernier. 


p  A  MICROMETER 
CALIPER  WITH  A  VERNIER 
j,  barrel  ;#,  thimble 


6  MEASURING  INSTRUMENTS 

Exercises.  Verniers  and  Micrometers 

1.  How  would  you   set  the   vernier  micrometer  in   the 
second  figure  on  page  5  to  read  0.4843"? 

2.  What  part  of   1"   is  indicated  by  the  sixteenth  line 
from  the  zero  mark  on  the  beam  of  a  vernier  caliper? 

3.  What  are  the  readings  on  the  beam  scale  and  on  the 
vernier  scale  of  a  vernier  caliper  for  a  thickness  of  J"? 

4.  How  would  you    set   a  vernier   micrometer   to   read 
0.5781"?  to  read  0.8704"? 

5.  What  are  the  readings  on  the  barrel,  the  thimble,  and 
the  vernier  of  a  ten-thousandths  micrometer  for  a  setting  of 
0.1093"?  for  a  setting  of  0.2789"? 

6.  How  far  does  the  screw  of  a  micrometer  move  at  each 
revolution  of  the  thimble  ? 

7.  How  would  you  set  a  vernier  micrometer  to  read  ^"  ? 

8.  A  machinist  takes  a  micrometer  caliper  which  is  fully 
closed  and  gives  the  thimble  three  revolutions.    What  is  the 
opening  of  the  caliper? 

9.  How  would  you  set  a  ten-thousandths  micrometer  to 
read  0.1878"  ?  to  read  0.6593"  ? 

10.  Through  what  part  of  a  revolution  must  the  thimble 
be  turned  to  move  the  screw  of  a  micrometer  caliper  0.004"  ? 
to  move  the  screw  0.017"? 

11.  Which,  if  either,  gives  the  more  accurate  measurement, 
a  vernier  caliper  or  a  micrometer  caliper?    State  fully  the 
reason  for  your  answer. 

12.  Find  the   distance  between  the   anvil  and  the  screw 
of  a  vernier  micrometer  when  the  reading  of  the  numbered 
graduations  on  the  barrel  is  4,  the  thimble  reading  is  6,  and 
the  vernier  reading  is  3. 


VERNIERS  AND  MICROMETERS 


13.  Find  the  distance  as  in  Ex.  12  when  the  reading  of 
the  numbered  graduations  on  the  barrel  is  7,  that  of  the  un- 
numbered graduations  is  3,  the  thimble  reading  is  4,  and  the 
vernier  reading  is  6. 

14.  Find  the  reading  of  a  micrometer  caliper  for  each  of 
the  settings  which  are  represented  in  the  following  figures: 


15.  In  the  following  table  let  A  represent  the  reading  of 
the  numbered  graduations  on  the  barrel  of  a  vernier  microm- 
eter, B  the  reading  of  the  unnumbered  graduations  on  the 
barrel,  C  the  thimble  reading,  and  D  the  vernier  reading.  Copy 
the  table  and  insert  the  number  representing  the  reading  of 
each  setting  in  the  blank  spaces  in  the  column  marked  E\ 


A 

P> 

C 

7) 

E 

6 

2 

19 

0 

1 

0 

6 

2 

9 

3 

1 

9 

7 

1 

20- 

4 

5 

1 

0 

7 

0 

0 

21 

1 

8 

3 

18 

6 

6 

0 

0 

3 

3 

3 

3 

3 

8 


MEASURING  INSTRUMENTS 


Gaging  the  Depth  of  a  Cut.  The  principle  of  the  micrometer 
screw  is  used  on  various  machines  in  gaging  the  depth  of  cut. 
The  picture  below  shows  a  graduated  collar  which  is  attached 
to  the  cross-feed  screw  of  a  lathe.  For  convenience  let 

x  =  the  number  of  inches  in  the 

endwise   movement   of    the 

feed  screw  for  each  rotation 

through  one  division  on  the 

collar ; 
N=  the  number  of  divisions  on  the 

collar ; 
L  =  the  number  of  inches  in  the 

lead  of  the  screw ;  that  is, 

the  distance  that  the  screw 

advances  in  a  single  turn. 
We  then  have  the  formula 
L 


GRADUATED  COLLAR  ON 
CROSS  FEED 


A,  cross  slide ;  B,  graduated 
collar ;  C',  cross-feed  screw 


Exercises.    Graduated  Feed  Screws 

1.  Find  the  distance  that  a  cross-slide  feed  screw  with  a 
lead  of  0.2"  moves  when  the  collar,  which  has  100  divisions, 
is  turned  through  five  divisions. 

2.  The  table  of  a  milling  machine  is  moved  by  a  double- 
thread  feed  screw  the  lead  of  which  is  •£•".    Into  how  many 
divisions  should  the  collar  of  the  feed  screw  be  divided  in 
order  that  the  table  shall  advance  0.001"  when  the  collar  is 
turned  through  one  division  ? 

3.  The  screw  that  raises  the  knee  of  a  milling  machine  is 
a  single-thread  screw  which   moves  the  knee   0.001"  when 
the  collar,  which  has  100  divisions,  is  turned  through  one 
division.    Find  the  lead  of  the  screw. 


CHAPTER  II 

SPEEDS  AND  FEEDS 

Cutting  Speed.  The  cutting  speed  of  a  machine  is  always 
given  in  feet  per  minute  (F. P.M.),  but  the  expression  has 
somewhat  different  meanings  for  different  machines. 

In  turning  work  on  a  lathe  it  means  the  number  of  linear 
feet,  measured  on  the  surface  of  the  work,  which  passes  the 
edge  of  the  cutting  tool  in  one  minute. 

On  a  planer  it  means  the  rate  in  F.P.M.  at  which  the 
work  passes  the  tool. 

On  a  shaper  it  means  the  rate  in  F.P.M.  at  which  the 
tool  passes  the  work. 

On  a  milling  machine  it  means  the  surface  speed  of  the 
cutter ;  that  is,  the  speed  of  a  point  on  the  rim  of  the  cutter. 

The  cutting  speed  is  not  always  the  same,  even  on  the 
same  machine,  but  depends  upon  the  following  conditions: 

1.  The  kind  and  quality  of  the  material. 

2.  The  kind  and  quality  of  the  cutting  tool. 

3.  The  depth  of  the  cut. 

4.  The  feed  of  the  tool,  that  is,  the  distance  traveled  side- 
ways by  the  tool  in  one  revolution  of  the  work  on  a  lathe,  or 
a  similar  distance  in  other  types  of  machines. 

5.  The  lubrication. 

Thus,  on  a  milling  machine  the  cutting  speed  may  be  30  F.P.M. 
for  steel,  50  F.P.M.  for  cast  iron,  and  90  F.P.M.  for  brass.  On  such  a 
machine  a  heavy  flow  of  oil  is  necessary  on  the  cutter  when  milling 
steel,  and  a  failure  to  provide  for  it  not  only  changes  the  speed  but 
materially  affects  the  life  of  the  cutter. 

9 


10 


SPEEDS  AND  FEEDS 


Cutting  Speeds  of  Lathes.  The  following  table  gives  the 
cutting  speed  in  F.P.M.  for  roughing  each  of  the  materials 
specified  with  lathe  tools  of  carbon  steel  or  of  high-speed  steel : 


MATERIAL 

CARBON  STEEL 

HIGH-SPEED  STEEL 

Annealed  tool  steel     . 

20 

45 

Machine  steel    .     .     . 

30 

65 

Wrought  iron    .     .     . 
Cast  iron  ..... 
Brass    

30 
40 

90 

65 
90 
190 

On  finishing  cuts  the  speed  is  50  %  greater  than  it  is  in  roughing. 

Letting  C  be  the  cutting  speed  in  F.P.M.,  R  the  number  of 
R.  P.  M.  of  the  work,  D  the  diameter  of  the  work  in  inches, 
and  taking  3£  as  the  value  of  TT,  we  have  these  formulas  : 


12  "  TTD 

The  factor  12  is  necessary  for  reducing  inches  to  feet. 

Exercises.   Cutting  Speeds  of  Lathes 

1.  If  a  tool  of  high-speed  steel  is  used,  at  how  many  R.  P.  M. 
should  a  cast-iron  pulley  18"  in  diameter  be  roughed? 

Use  the  formula  for  R,  cancel,  and  give  the  result  to  the  nearest  unit. 

2.  Find  the  speed  of  the  finishing  cut  in  boring  the  hub 
of  a  cast-iron  pulley  to  a  diameter  of  Zf~'  with  a  carbon- 
steel  tool;  with  a  high-speed  steel  tool. 

3.  Find  the  speed  at  which  a  lathe  should  run  to  rough  a 
brass  bushing  1|"  in  diameter  with  a  high-speed  steel  tool. 

4.  Some  machine-steel  axles  are  roughed  with  a  carbon-steel 
tool  in  an  axle  lathe  which  has  a  fixed  speed  of  75  R.P.M. 
What  is  the  diameter  of  each  axle  ? 


PLANERS  11 

Cutting  Speeds  of  Planers.  Planers  are  usually  so  belted 
as  to  give  only  one  cutting  speed.  This  speed  is  usually 
about  25  F.P.M.  and  is  used  for  brass,  cast  iron,  and  steel. 
Only  the  direct  stroke  is  a  cutting  stroke,  the  return  stroke 
being  more  rapid,  usually  two  or  three  times  as  rapid  as  the 
cutting  stroke.  It  is  customary  to  say  that  the  return  stroke 
is  then  2  to  1  or  3  to  1 ;  and  to  speak  of  the  planer  as  a 
2-to-l,  or  a  3-to-l,  planer. 

In  setting  up  the  work  on  a  planer  a  distance  of  V  is 
always  allowed  at  each  end  of  the  work  for  the  tool  to  clear. 

Exercises.   Cutting  Speeds  of  Planers 

1.  How  long  will  it  take  to  make  one  cut  the  length  of  a 
cast-iron  block  14'  10"  long,  the  cutting  speed  of  the  planer 
being  25  F.P.M.  and  the  return  stroke  being  3  to  1  ? 

Allowing  V  at  each  end  of  the  work  for  the  tool  to  clear,  the  total 
forward  movement  is  15'.  Hence  the  forward  movement  will  take  ^£, 
or  |,  of  a  minute.  How  long  will  it  take  to  make  the  cut? 

2.  A  planer  to  be  used  for  planing  cast  iron  is  belted  to 
give  a  cutting  speed  of  45  F.P.M.    If  the  return  speed  is 
90  F.P.M.,  how  many  strokes  per  minute  will  the  planer 
make  when  the  length  of  the  stroke  is  2'  6"?  when  it  is  6'  ? 
when  it  is  8'  9"  ?   when  it  is  12'  6"  ? 

3.  If  a  2-to-l  planer  makes  six  strokes  per  minute  and 
the  length  of  each  stroke  is  3'  4",  what  is  the  cutting  speed  ? 

4.  How  long  will  it  take  to  make  36  cuts  on  an  iron  cast- 
ing 43"  long,  the  cutting  speed  ,of  the  planer  being  45  F.  P.  M. 
and  the  return  stroke  being  2  to  1  ? 

5.  On  a  casting  1'  7fr  long  a  3-to-l  planer  is  making  12 
strokes  per  minute.    What  is  the  cutting  speed  ? 

6.  In  Ex.  5  how  many  strokes  per  minute  would  be  re- 
quired to  have  the  same  cutting  speed  on  a  2-to-l  planer? 


12  SPEEDS  AND  FEEDS 

Cutting  Speeds  of  Milling  Machines.  The  table  below  gives 
the  cutting  speeds  in  F.P.M.  for  milling  each  of  the  following 
materials  on  a  milling  machine  with  carbon-steel  cutters : 


Annealed  tool  steel 
Machine  steel 

30 
40 

Cast  iron 
Brass 

45 

90 

For  high-speed  steel  cutters  the  speeds  are  100%  higher. 

In  finding  the  cutting  speeds  of  milling  cutters  we  use  the 
formulas  given  on  page  10,  letting  D  be  the  diameter  of  the 
cutter  in  inches  instead  of  the  diameter  of  the  work. 


Exercises.   Cutting  Speeds  of  Milling  Machines 

1.  Find  the  cutting  speed  of  a  side-facing  cutter  6"  in 
diameter  running  at  30  R.P.M. 

Use  the  formula  for  C  given  on  page  10,  and  give  the  result  to  the 
nearest  unit.    Use  cancellation  whenever  possible. 

2.  Find  the  cutting  speed  of  a  milling  cutter  1J"  in  diam- 
eter running  at  85  R.P.M. 

3.  Find  the  number  of  R.P.M.  for  milling  a  brass  casting 
with  a  high-speed  steel  cutter  4"  in  diameter. 

4.  Find    the   cutting   speed   of   a   milling    cutter    2f"   in 
diameter  running  at  55  R.P.M. 

5.  How  many  R.P.M.  should  a  carbon-steel  cutter  3^"  in 
diameter  make  in  milling  a  square  on  a  machine-steel  shaft  ? 

6.  Babbitt   metal   can   be   milled   at  a  cutting   speed   of 
120  F.P.M.     Find   the  number  of   R.P.M.   for   a  41-inch 
cutter  when  milling  a  piece  of  this  metal. 

7.  Using  a  carbon-steel  cutter  5£"  in  diameter,  find  the 
number  of  R.P.M.  for  milling  a  cast-iron  lathe  bed. 


DRILLS  13 

Speeds  of  Drills.  In  the  case  of  drills  the  number  of  R.P.M. 
for  the  drill  is  found  by  dividing  a  certain  number  called  a 
constant,  because  it  is  always  the  same  for  the  given  material, 
by  the  diameter  D  of  the  drill.  For  example  : 

For  machine  steel,     R.  P.  M.  =  —  • 

125 
For  cast  iron,  R.  P.  M.  =  —  • 


For  brass,  R.P.M.  =--- 

In  these  cases  the  constants  are  100,  125,  and  225  respectively. 

The  depth  of  cut  for  small  drills  is  usually  from  0.002" 
to  0.005"  for  each  revolution,  while  for  large  drills  it  is  from 
0.005"  to  0.020". 

Exercises.    Speeds  of  Drills 

1.  Find  the  number  of  R.P.M.  for  a  J-inch  drill  when 
drilling  a  cast-iron  lathe  bed. 

2.  How  fast  should  you  run  a  drill  J^"  in  diameter  when 
drilling  brass  bearings  ? 

3.  When   centering  a  machine-steel  shaft  with  a  ^-inch 
drill,  how  fast  should  the  drill  rotate  ? 

4.  If  a  lathe  has  a  fixed  speed  of  250  R.P.M.,  what  is  the 
largest  diameter  of  drill  that  should  be  used  on  cast  iron  ? 

5.  Find   the  number   of   R.P.M.  for  a  #30   drill   when 
drilling  machine  steel.       f 

A  #30  drill  has  a  diameter  of  0.1285". 

6.  Find  the  number  of  R.P.M.  for  a  lT3g-inch  drill  when 
drilling  cast-iron  surface  plates. 

7.  Find  to  the  nearest  g\"  the  largest  diameter  of  drill  that 
should  be  used  in  drilling  brass  at  350  R.P.M. 


14  SPEEDS  AND  FEEDS 

Exercises.    Review 

1.  Find  the  number  of  R.P.M.  of  a  lathe  for  roughing 
the  outside   of  the  brass  cylinder,  which  is   shown  in  the 
blueprint  on  page  15,  with  a  tool  of  carbon  steel. 

2.  Find  the  speed  of  the  finishing  cut  in  boring  the  hole 
in  the  same  cylinder  with  a  high-speed  tool. 

3.  Find  the  number  of  R.P.M.  for  drilling  the  center 
hole  in  the  spur-gear  blank. 

4.  Using  a  carbon-steel  tool,  find  the  number  of  R.P.M. 
for  roughing  the  face  R  of  the  spur-gear  blank. 

5.  If  the  spur-gear  blank  in  Ex.  4  were  made  of  machine 
steel  instead  of  cast  iron  and  if  the  tool  were  of  high-speed 
steel,  should  the  speed  of  the  lathe  be  increased  or  should 
it  be  decreased  when  making  the  cut?    How  much  should 
be  the  increase  or  decrease? 

6.  What  should  be  the  number  of  R.P.M.  for  drilling  the 
hole  in  the  machine-steel  wheel  shown  in  the  blueprint  ? 

7.  Find  the  number  of  R.P.M.  for  a  finishing  cut  over 
the  outside  of  the  steel  wheel  with  a  tool  of  carbon  steel. 

8.  Find  the  number  of  R.P.M.  for  drilling  the  l^g-inch 
hole  in  the  cast-iron  piston. 

9.  Using  a  high-speed  tool,  find  the. number  of  R.P.M.  for 
roughing  the  outside  of  the  piston. 

10.  Using  a  carbon-steel  tool,  find  the  number  of  R.P.M. 
for  finishing  the  piston-ring  grooves  shown  in  the  blueprint. 

11.  Find  the  number  of  R.P.M.  for  milling  a  brass  casting 
with  a  high-speed  cutter  7J"  in  diameter. 

12.  If  a  drill  press  is  belted  to  make  365  R.P.M.,  find  to 
the  nearest  fa"  the  largest  diameter  of  drill  that  can  be  used 
when  drilling  cast  iron. 


REVIEW  EXERCISES 


15 


16  SPEEDS  AND  FEEDS 

Cutting  Speed  of  Shapers.  There  are  two  leading  kinds  of 
shapers,  namely,  geared  shapers  and  crank  shapers. 

The  cutting  speed  of  a  geared  shaper  is  found  in  the  same 
way  as  that  of  a  planer,  as  explained  on  page  11. 

In  the  case  of  a  crank  shaper,  however,  the  number  of 
strokes  per  minute  depends  upon  the  speed  of  the  cone 
pulley  which  drives  the  shaper.  In  this  kind  of  shaper  the 
return  stroke  has  about  twice  the  speed  of  the  forward 
stroke ;  that  is,  it  is  said  to  be  about  2  to  1. 

For  example,  if  a  crank  shaper  is  making  50  R.  P.  M.,  if  the  length 
of  the  stroke  is  I/,  and  if  the  return  stroke  is  twice  as  rapid  as  the 
forward  stroke,  what  is  the  cutting  speed  of  the  shaper  ? 

Since  the  return  stroke  is  twice  as  fast  as  the  forward  one,  it  takes 
half  as  long  to  make  it.  Hence  the  forward  stroke  takes  §  and  the  re- 
turn stroke  takes  ^  of  the  time  required  for  one  revolution. 

In  1  min.  there  are  50  forward  strokes  of  V  each  and  50  return 
strokes  of  the  same  length,  the  50  forward  strokes  taking  §  of  a 
minute  and  the  50  return  strokes  taking  ^  of  a  minute. 

Hence  the  cutting  speed  is  50'  in  §  min.,  or  f  of  50'  in  1  min. 

That  is,  the  cutting  speed  is  75  F.P.M. 

Exercises.    Cutting  Speeds  of  Shapers 

1.  The  ram  of  a  crank  shaper  makes  85  strokes  per  minute, 
and  the  length  of  the  cut  is  2".    Find  the  cutting  speed. 

In  problems  dealing  with  shapers  take  the  ratio  of  speeds  as  2  to  1, 
as  given  above,  unless  otherwise  stated. 

2.  A  crank  shaper  is  cutting  at  the  rate  of  40  F.P.M.  on  a 
14-inch  cut.    Find  the  number  of  R.P.M. 

3.  If  a  crank  shaper  makes  18  R.P.'M.  and  the  length  of 
the  cut  is  9",  what  is  the  cutting  speed  ? 

4.  If  a  crank   shaper  is  cutting  a  4-inch  casting  at  the 
rate  of  50  F.P.M.,  what  is  the  number  of  R.P.M.? 

5.  Solve  Ex.  4  for  a  cutting  speed  of  40  F.P.M. 


FEEDS 


17 


Expression  of  Feeds.    Feeds  can  be  expressed  in  several 
different  ways,  the  following  being  the  most  common: 

1.  By  the  number  of  revolutions  which  the  work  makes 
while  the  tool  is  advancing  1". 

That  is,  we  may  speak  of  a  feed  as  75  revolutions  to  the  inch. 

2.  By  the  number  of  inches  that  the  tool  advances  along 
the  work  at  each  revolution  or  stroke. 

In  the  above  case  we  may  express  the  feed  as  J^"  per  revolution. 

Standard  Feeds.    The  following  are  certain  standard  feeds 
in  common  use,  each  being  expressed  in  inches  per  revolution : 


MATERIAL 

LATHE 

PLANER 

MILLING 
MACHINE 

DRILLS 

Annealed  tool 
steel 

A  to  A 

«V<°1 

0.002  to  0.015 

0.002  to  0.015 

Machine  steel  or 
wrought  iron 

AM 

«VtoJ 

0.005  to  0.025 

0.002  to  0.015 

Cast  iron 

A  to  A 

iVtoi 

0.005  to  0.075 

0.005  to  0.020 

Brass 

A  to  A 

A  toj 

0.002  to  0.125 

0.005  to  0.020 

Exercises.    Feeds 

1.  A  milling  cutter  revolves  85  times  while  the  table  is 
moving  V1.    What  is  the  feed  in  inches  per  revolution? 

2.  In  1  min.  a  lathe  tool  moves  1J"  while  the  work  makes 
250  revolutions.    What  is  the  feed  in  revolutions  per  inch? 

3.  A  shaper  makes  15  strokes  while  the  table  moves  1J". 
What  is  the  feed  in  inches  per  stroke  ? 

4.  A  drill  making  244  R.P.M.  has  a  feed  of  0.008".    How 
long  will  it  take  it  to  drill  through  a  plate  2J-"  thick  ? 


18  SPEEDS  AND  FEEDS 

Exercises.    Review 

1.  The  bench  block  shown  in  the  blueprint  on  page  19  is 
made  of  cast  iron  and  fy1!  was  left  on  the  surfaces  marked 
/  for  finishing  to  the  dimensions  given.    If  a  roughing  cut 
is  taken  with  a  1-inch  feed  and  a  finishing  cut  with  a  ^-inch 
feed,  how  long  will  it  take  to  plane  the  top  of  the  casting 
on  a  3-to-l  planer  with  a  cutting  speed  of  25  F.P.M.? 

In  such  problems  find  the  results  to  minutes,  calling  any  fraction  a 
whole  minute,  unless  otherwise  directed. 

2.  The   die-shoe  dimensions  given    in    the    blueprint   are 
for  the  rough  machine-steel  casting.    Taking  one  cut  with  a 
0.015-inch  feed  and  using  a  carbon-steel  side-milling  cutter 
having  a  3-inch  face  and  a  diameter  of  4",  how  long  will  it 
take  to  machine  the  entire  surface  of  the  shoe  ? 

It  will  require  six  cuts  on  the  top  of  the  shoe,  five  on  the  bottom, 
one  at  each  end,  and  one  on  each  long  side. 

3.  The  multiple-spindle  drill  table  is  made  of  cast  iron. 
Find  the  time  required  to  plane  the  top  on  a  2-to-l  planer 
with  a  cutting  speed  of  25  F.  P.  M.,  giving  it  one  roughing  cut 
with  a  ^g-inch  feed  and  one  finishing  cut  with  a  j^-inch  feed. 

4.  Using  a  2-to-l  planer  with  a  cutting  speed  of  25  F.P.M. 
and  a  ^g-inch  feed,  how  long  will  it  take  to  make  one  cut  over 
the  two  surfaces  marked  /  on  the  machine-steel  angle  guide  ? 

5.  How  many  R.P.M.  should  be  made  by  a  carbon-steel 
T-slot  cutter  1-J."  in  diameter  when  milling  the  T-slots  in  the 
cast-iron  milling-machine  table  ? 

6.  What  should  be  the  speed  of  a  4-inch  high-speed  cutter 
when  milling  the  4-inch  groove  in  the  milling-machine  table  ? 

7.  Calculate  the  time  required  to  take  a  single  cut  over 
the  top  of  the  milling-machine  table,  using  a  ^V-kich  feed 
on  a  3-to-l  planer  with  a  cutting  speed  of  25  F.P.M. 


REVIEW  EXERCISES 


19 


20  SPEEDS  AND  FEEDS 

8.  Find  the  time  required  to  drill  through   a  cast-iron 
plate  lTy  thick  with  a  {-inch  drill  and  a  feed  of  0.008". 

In  this  case  give  the  result  to  the  nearest  second. 

9.  A  machine-steel  cylinder  4'  9"  long  is  to  be  turned  in 
a  lathe  to  a  diameter  of  6|".  Using  a  ^-inch  feed  and  a  high- 
speed tool,  how  long  will  it  take  to  make  the  roughing  cut  ? 

10.  In  Ex.  9,  using  a  ^-inch  feed,  how  long  will  it  take  to 
make  the  finishing  cut? 

11.  Using  a  carbon-steel  cutter,  how  many  R.P.M.  should 
a  y9g-inch  end  mill  make  while  cutting  the  slots  in  a  cast-iron 
milling-machine  table  ? 

12.  Using  a  J^-inch  feed  and  a  high-speed  tool,  how  long 
will  it  take  for  a  roughing  cut  over  the  face  of  a  cast-iron 
pulley  which  has  a  diameter  of  28"  and  a  face  15"  wide  ? 

13.  If  a   2-to-l   planer  makes  eight  strokes  of   5'  6"  per 
minute,  what  is  the  cutting  speed  of  the  planer  ? 

14.  Some  machine-steel  axles  9'  2"  long  are  turned  down  to 
3 1"  in  diameter.   If  two  cuts  are  made  with  a  high-speed  tool, 
the  roughing  cut  with  a  J^-inch  feed  and  the  finishing  cut  with 
a  ^L-inch  feed,  how  long. should  it  take  to  finish  75  axles? 

First  find  the  time  to  the  next  full  minute  for  each  axle. 

15.  How  many  R.P.M.  must  a  2-to-l  crank  shaper  make 
to  attain  a  cutting  speed  of  35  F.P.M.  on  a  7J-inch  casting? 

16.  How  much  time  is  needed  to  make  a  single  cut  with 
a  J^-inch  feed  over  one  side  of  an  iron  casting  22"  wide  and 
8'  4"  long  on  a  planer  whose  forward  speed  is  35  F.P.M. 
and  whose  return  speed  is  60  F.P.M.? 

17.  Find   the  number  of  R.P.M.  for  drilling  a  cast-iron 
block  |"  thick  with  a  drill  J"  in  diameter.    Find  the  feed 
necessary  to  drill  through  the  block  in  25  sec. 


CHAPTER  III 

TAPERS  AND  TAPER  TURNING 

Taper.    The  difference  in  diameter  of  a  piece  for  a  unit  of 
length  is  called  the  taper  of  the  piece. 

The  piece  may  be  round  like  a  lathe  center  or  flat  like  a  taper  gib. 
The  taper  is  usually  stated  by  giving  the  difference  in  diameter  in 
inches  for  a  foot  of  length,  as,  for  example,  \"  per  foot. 

Common  Tapers.    The  tapers  most  frequently  used  are: 
Morse :  approximately  0.625"  per  foot  (all  sizes) 
Brown  and  Sharpe:  0.500"  per  foot  (all  except  #10) 
Jarno :  0.600"  per  foot  (all  sizes) 

Symbols.   The  following  symbols  or  abbreviations  are  used 
in  connection  with  tapers: 

T.P.I.  =  taper  per  inch  D  =  larger  diameter 

T.P.F.  =  taper  per  foot  d  =  smaller  diameter 

T.P.L.  =  taper  in  any  length         1  =  length  of  the  taper 

All  the  measurements  are  always  stated  in  inches. 

Formulas.    The  following  formulas  may  be  referred  to  in 
solving  the  exercises  on  tapers: 

T.P.F.  //XT.  P.  F 

T.P.I.  =  _  D 


12(D-d)  //XT.P.FA 

V        12        / 


T.P.F.  =  —^- d 

7xT.P.F.  12  (D-d) 

T.P.L.  =  ___  '=TRFT 

21 


22 


TAPERS  AND  TAPER  TURNING 


Exercises.    Tapers 

1.  Find  the  T.P.F.  of  the  taper  collar  shown  in  the  blue- 
print on  page  23. 

2.  If  the  taper  of  the  end  mill  is  0.625"  per  foot,  what  is 
the  length  of  the  tapered  part  ? 

3.  If  the  T.P.F.  of  the  milling-machine  arbor  is  0.500", 
find  the  diameter  of  the  larger  end  of  the  taper. 

4.  The  T.P.F.  of  the  lathe  center  is  T9g".  Find  the  length  A. 

5.  Find  the  T.P.F.  of  the  flat  drill  shown  in  the  blueprint. 

6.  Find  the  T.P.F.  of  a  boring  bar  which  is  tapered  16" 
of  its  length  to  diameters  of  1|"  and  J"  respectively. 

7.  The   shank    of   a   drill   is   31"    long,    D=  0.421",    and 
d  =  0.375".    Find  the  T.P.F. 

8.  A  drill  collet  has  a  T.P.F.  of  0.602"  and  d  =  4".    Find 

o 

the  diameter  at  a  point  3|"  from  the  smaller  end. 

9.  The  following  formulas,  in  which  N  is  the  number  of  the 
taper  in  the  table,  D  is  the  diameter  of  the  larger  end  in 
inches,  d  is  the  diameter  of  the  smaller  end  in  inches,  and  I  is 
the  length  of  the  taper  in  inches,  are  used  in  a  J  arno  taper  : 


Copy  the  following  table  and  fill  the  columns,  using  decimals 


N 

D 

d 

I 

N 

/; 

d 

/ 

N 

j) 

<l 

i 

1 

7 

13 

2 

8 

14 

3 

9 

15 

4 

10 

16 

5 

11 

17 

6 

12 

18 

TAPERS 


23 


24 


TAPEKS  AND  TAPER  TURNING 


Taper  Turning.    There  are  three  ways  of  turning  tapers: 
1.  By  offsetting  the  tailstock,  that  is,  by  moving  the  two 
centers  out  of  alignment  by  means  of  the  screws  S,  as  shown. 


OFFSET   TAILSTOCK 

A-B,  center  line  of  the  lathe  ;  S,  S,  screws  for  moving  tailstock  ;  T,  tailstock  ; 
O,  offset  of  tailstock  center 

To  find  0,  the  amount  of  offset,  these  formulas  are  used* 

a.  When  the  taper  runs  the  entire  length  of  a  bar, 

0  =  \(D-d). 
For  example,  if  a  bar  is  to  be  turned  taper  to  diameters  of  1^"  and  1", 

o  =  KD  -  d)  =  iar  -  1")  =  i  x  r  =  i". 

b.  When  the  taper  runs  only  part  of  the  length  of  a  bar, 


21 

where  L  is  the  total  length  of  the  bar  in  inches. 

For  example,  if  a  taper  6"  long  with  diameters  of  2^"  and  2"  respec- 
tively is  to  be  turned  on  a  bar  18"  long, 

0  =  (D  ~  O  L  -  (2?  ~  2)  x  18//  =       18"       =3" 
21  2x6          ~2x2x6~4* 

c.  When  part  of  a  bar  is  to  be  tapered  to  a  given  T.P.F., 


For  example,  if  a  T.P.F.  of  0.6"  is  to  be  turned  on  a  bar  12"  long, 
0  =  2^  (T.P.F.  x  L)  =  ^  x  0.6  x  12"  =  0.3". 


,-'         '£V 

rUTTT>XFTArr«  •  

a 

/ 

TAPER 

2.  By  using  the  taper  attachment  shown  below,   which 
causes  the  tool  to  feed  transversely  at  the  same  time  that  it 


TAPER   ATTACHMENT 

The  graduations  on  the  ends  E,  E  are  in  T.P.F.,  and  C  is  attached  to  the  cross 

slide  of  the  lathe.   The  graduations  at  one  end  are  usually  in  tenths  and  at  the 

other  end  in  eighths 

feeds  longitudinally,  thus  turning  a  taper.  This  attachment 
can  be  used  for  turning  either  ex- 
ternal or  internal  tapers.  In  using 
a  taper  attachment  neither  the  tail- 
stock  offset  nor  the  distance  between 
centers  need  be  considered. 

3.  By  using  the  compound  rest, 
one  form  of  which  is  here  shown. 
Since  the  tool  can  be  set  at  any 


COMPOUND    REST 

The  base  G  is  graduated  in 
degrees 


desired  angle,  very  steep  tapers  can  be  cut  with  the  compound 


METHODS   OF   GRADUATING   THE   BASE   OF   A   COMPOUND  REST 
The  line  A-B  represents  the  center  line  of  the  lathe 

rest.    The  base  of  the  rest  is  usually  graduated  in  degrees  in 
one  of  the  four  ways  shown  in  the  figure  above. 

The  method  of  setting  the  compound  rest  is  described  on  page  28. 


26 


TAPERS  AND  TAPER  TURNING 


Exercises.    Offset  Method 

1.  Find  the  T.P.F.  of  the  drill-holder  handle  shown  in 
the  blueprint  on  page  27. 

2.  In  Ex.  1  how  far  should  the  tailstock  center  be  offset 
in  order  to  turn  the  handle  to  the  required  taper  ? 

3.  The  crowned  pulley  is  to  be  turned  on  a  10-inch  arbor. 
What  distance  should  the  tailstock  center  be  offset? 

4.  Find  the  T.P.F.  of  the  crowned  pulley. 

5.  If  the  thimble  is  to  be  turned  on  a  6J-inch  mandrel, 
find  the  T.P.F.  and  the  distance  which  the  tailstock  center 
should  be  offset  in  turning  the  taper. 

6.  In  turning  the  taper  reamer,  what  should  be  the  offset 
of  the  tailstock  center?    What  is  the  T.P.F.? 

7.  In  turning  the  taper  spindle,  what  should  be  the  offset 
of  the  tailstock  center? 

8.  A  taper  15"  long  with  diameters  of  1.234"  and  0.984" 
respectively  is  to  be  turned  on  a  shaft  17"  long.    Find  the 
offset  for  the  tailstock  center. 

9.  Copy  the  following  table  of  measurements  of  Morse  stand- 
ard taper  reamers  and  supply  the  missing  numbers : 


No. 

D 

d 

I 

T.P.F. 

0 
1 

2 

0.369" 
0.741 

0.252" 
0.369 

21" 
2i| 

0.600" 
0.602 

3 

0.979 

0.778 

0.602 

4 

1.280 

1.020 

0.623 

5 

1.790 

1.475 

6 

6 

2.559 

2.116 

8J 

7 

2.750 

12 

0.625 

OFFSET  METHOD 


27 


28  TAPERS  AND  TAPER  TURNING 

Angles.  A  steep  taper  is  usually  referred  to  as  an  angle, 
and  in  designating  the  angle  we  generally  give  either  the 
included  angle  or  the  angle  with  the  center  axis.  In  the  figure 
below  the  angle  DAC  is  the  included  angle  and  the  angle  a 
is  the  angle  with  the  center  axis.  The  following  formula  is 
used  in  finding  a,  the  angle  with  the  center  axis : 

D-d 

—  --- 

The  included  angle  is  found  by  multiplying  a  by  2. 
For  example,  if  D  =  O.S75",  d  =  0.250",  and  I  =  1£",  we  have 

tana  =  ^7  =  |-=1  =  J  x  £  x  §  =  1  =  0.2500. 

From  the  table  on  page  146  we  find  that,  to  the  nearest  minute, 
0.2500  =  tan  14°  2'. 

Therefore  a,  the  angle  with  the  center  axis,  is  14°  2',  and  2  a,  the 
included  angle,  is  28°  4'. 

If  the  taper  per  foot  is  given,  the  above  formula  becomes 

T.P.F 


tan  a  = 


24 


Setting  the  Compound  Rest.  To  turn  a  given  angle  the 
compound  rest  shown  on  page  25  is  swiveled  from  its  zero 
position,  which  is  perpendicular  to 
the  center  line  of  the  lathe,  so 
that  the  axis  of  the  feed  screw  of 
the  rest  shall  be  parallel  to  the 
side  of  the  angle.  Thus,  to  turn 
angle  a  in  this  figure  the  rest  is 
swiveled  so  that  the  graduated  base  D 

is  set  at  the  complement  of  a ;  that  is,  at  90°  —  30°,  or  60°. 
To  turn  angle  £>,  which  is  measured  from  the  perpendicular 
to  the  center  axis,  the  rest  is  set  direct ;  that  is,  at  60°. 


COMPOUND  REST 


29 


Exercises.    Use  of  the  Compound  Rest 
Find  the  included  angles  for  the  following  tapers  per  foot  : 
1.  0.500".  2.  0.602".  3.  T9g".  4.  0.750". 

5.  A  drill  collet  is  to  be  bored  taper  by  the  aid  of  a 
compound  rest.  If  it  is  required  that  D  —  1  Jg  ",  d  =  1|",  and 
I  =  4J",  find  the  angle  made  with  the  center  axis. 

Find  the  angles  made  with  the  center  axis  in  these  tapers : 

6T) Ql"     /7 11"    7 1  H        «       T)        91  It     fl        11^7        Q" 
.  JJ  —  t>2  ,  a  —  ±^  ,  t  —  i  .     o.  JL*  =  z>±  ,  a  =  ig-  ,  6  =  o  . 

10.  In  turning  the  tapers  A  and  B  on  the  taper  bearing 
shown  in  this  figure,  at  what  angles  should  the  compound 
rest  be  set? 

B 

First  find  the  angle  which    _ 
each    taper    makes    with    the 
center  axis  of  the  bearing. 

11.  Find  the  T.  P.F.  of  taper  A  on  the  bevel-gear  blank 
here   shown.    At  what   angle   should  the 

L,     1 1//  J 

compound  rest  be  set  to  cut  the  taper? 

12.  If    in    a   taper    socket   D  =  0.435", 
d  =  0.300",   and   I  =  2f ",    find    the    angle 
made  with  the  axis. 

13.  If    in    the    taper    shank    of    a   drill 
D  =  0.475",  d=  0.388",  and  Z=lf",  find 
the  included  angle. 

14.  A  counter  bore  has  a  taper  shank  in  which  D  =  0.416", 
d  =  0.312",  and  Z  =  lf".    Find  the  included  angle. 

15.  A  bevel-gear  blank  is  to  be  turned  to  an  included  angle 
of  134°.    Find  the  T.P.F.  of  the  blank  and   the  angle  at 
which  to  set  the  compound  rest. 


53" 


"T 


3" 


1 


30  TAPERS  AND  TAPER  TURNING 

Exercises.    Review 

1.  The  center  punch  shown  in  the  blueprint  on  page  31 
is  to  be  turned  in  a  chuck  with  the  aid  of  a  compound  rest. 
Find  the  angle  which  each  taper  makes  with  the  axis. 

2.  The  rotary  oiler  shown  in  the  blueprint  is  used  to  oil 
the  ways  of  a  planer.    Find  the  included  angle. 

3.  In  the  drill   socket   shown  in  the  blueprint   find  the 
T.P.F.  and  the  angle  made  with  the  axis. 

4.  In  the  speed-lathe  center  shown  in  the  blueprint  find 
the  T.P.F.  and  the  included  angle. 

5.  A  milling-machine  arbor  18"  long  is  to  be  tapered  4£" 
on  one  end  to  0.500"  per  foot.     Find  the  offset  of  the  tail- 
stock  center  for  turning  the  required  taper. 

6.  A  reamer  has  a  T.P.F.  of  0.625"  and  its  diameters  are 
0.500"  and  0.750"  respectively.    Find  I 

7.  If  the  maximum  offset  of  the  tailstock  center  of  a  lathe 
is  iy,  what  is  the  greatest  T.P.F.  that  can  be  cut  on  a 
shaft  6"  long  ?  on  a  shaft  2'  3"  long  ?  on  a  shaft  3'  6"  long  ? 

8.  A  taper  bushing  has  diameters  of  2|"  and  1|",  respec- 
tively, and  a  T.P.F.  of  0.500".    Find  I 

9.  In  the  taper  shank  of  a  cutter  which  has  a  T.P.F.  of 
0.600",  d  =  If"  and  I  =  3|".    Find  2). 

In  each  of  the  following  find  the  angle  made  with  the  axis : 

10.  D  =  0.354",  d  =  0.279",  and  1=  3f". 

11.  D  =  1.125",  d=  0.900",  and  Z  =  4i". 

12.  />  =  1.231",  d  =  1.020",  and  Z  =  4Ty. 

13.  I>=  2.494",  d=  2.116",  and  l  =  l\". 

14.  D=  1.2888",  d  =  1.0446",  and  /=5jJ". 

15.  D  =  3.976",  d  =  2.165",  and  I  =  3|". 


REVIEW  EXERCISES 


31 


32 


TAPERS  AND  TAPER  TURNING 


Measuring  Tapers  with  a  Sine  Bar.    An  instrument  known 
as  a  sine  bar  is  often  used  to  measure  the  angle  of  a  taper. 


MEASURING    TAPERS   WITH   A   SINE    BAR 

P,  scraped  surface  plate ;  7?,  R,  plugs ;  S,  hardened-steel  sine  bar ; 
T,  taper  plug  gage ;   U,  straight  edge ;   V,  vernier  height  gage 

The  taper  to  be  measured  is  placed  on  the  straight  edge  U, 
which  is  parallel  to  the  surface  plate  P,  and  the  sine  bar  S, 
which  has  two  plugs  .#,  R  set  10"  apart,  is  clamped  along  the 
taper.  Then  r,  the  difference  in  height  in  inches  between 
the  plugs,  is  found  by  means  of  the  height  gage  F.  Letting 
A  be  the  included  angle,  we  have  the  following  formulas: 


sinA  =  — 
10 


r  =  10  sin  A 


For  example,  in  the  above  figure  r  =  0.525",  and  we  have 

v          0  r»9'i 

sin  .4  =  —  =  —^-  =  0.0525  ;  whence  A  =  3°  V. 
Therefore  the  included  angle  of  the  taper  plug  gage  is  3°  1'. 

Testing  Tapers.  To  test  a  taper  for  a  given  angle,  the 
difference  in  height  r  of  the  plugs  is  found  from  the  second 
formula,  and  bar  S  is  set  to  this  distance  by  means  of  the 
height  gage.  The  taper  is  then  tested  between  bars  S  and  U. 

For  example,  what  should  be  the  difference  in  height  of  the  plugs 
for  testing  a  taper  which  is  to  have  an  included  angle  of  26°  30'? 

We  have  r  =  10  sin  A  =  10  x  0.4462  =  4.462. 

Hence  the  difference  in  height  of  the  plugs  should  be  4.462". 


MEASURING  TAPERS 


33 


Measuring  Tapers  with  Disks.    The  angle  of  a  taper  may 
also  be  measured  by  means  of  two  disks  of  unequal  diameters. 


1 


MEASURING    TAPERS   WITH    DISKS 

7?,  J?,  hardened-steel  edges ;  D,  d,  disks  of  different  diameters ;  I,  distance 
between  centers  of  disks 

The  disks  are  placed  as  shown  above,  and  the  straight 
edges  R,  R,  which  are  made  of  hardened  steel  and  carefully 
ground,  are  adjusted  so  that  the  tangent  lines  form  the  taper. 

Taking  a  as  the  angle  with  the  center  axis,  D  as  the 
larger  diameter,  d  as  the  smaller  diameter,  and  I  as  the  dis- 
tance between  the  centers,  as  shown  in  the  figure  below, 
we  have 


sin  a  = 


whence 


sin  a  = 


I 

D-d 

21 


Angle  a  can  then  be  found  from  a  table  of  sines,  and 
from  it  we  can  find  2  a,  the  included  angle  of  the  taper. 
Furthermore,  from  the  formula  for  sin  a  we  have 


so  that,  given  D,  d,  and  the  angle  with  the  axis,  we  can  find  L 


34 


TAPERS  AND  TAPER  TURNING 


Exercises.    Measuring  Tapers 

Find  the  difference  in  height  of  the  plugs  for  setting  a  sine 
bar  to  test  each  of  the  following  tapers  per  foot : 

1.  0.600".       3.   0.623".        5.   f".          7.  1^".         9.  2£". 

2.  0.602".       4.  0.630".       6.  £".       8.  If".         10.  2|". 

11.  If  the  disk  diameters  are  1"  and  2"  respectively  and 
if  Z,  the  center  distance,  is  4",  what  is  the  included  angle 
of  the  taper? 

12.  If  the  disk  diameters  are  1"  and  1.5"  respectively  and 
if   Z  =  4",  what    is   the  T.P.F.?    the    included  angle?    the 
angle  formed  with  the  axis  ? 

13.  If  the  taper  is  0.5"  per  foot  and  the  disk  diameters  are 
0.75"  and  1"  respectively,  what  is  I? 

14.  The  figure  and  table  below  show  certain  dimensions  of 
Morse  standard  tapers, 

all  the  measurements 
being  given  in  inches. 
Copy  the  table  and  fill 
in  the  blank  columns 
marked  Z>,  c?,  and 
"  Angles  with  Axis." 


T.P.F.  OF 
OUTER 
HOLE 

T.P.F.  OF 
INNER 
HOLE 

A 

B 

c 

E 

D 

d 

ANGLES 

WITH 

Axis 

0.602 

0.600 

31 

2J_ 

0.938 

0.475 

.623 
.630 
.626 

.602 

.602 
.623 

H 

5I 

sf 

1.231 
1.748 
2.494 

0.700 
0.938 
1.231 

.626 

.630 

n 

5i 

2.494 

1.748 

CHAPTER  IV 


SCREW  THREADS 

Thread.  When  a  uniform  spiral  groove  is  cut  around  a 
cylindric  bar,  the  material  left  by  the  cutting  tool  forms  a 
projection  which  is  called  a  thread,  or  a  screw  thread. 

The  diameter  of  the  cylindric  bar  on  which  the  screw 
thread  is  cut  is  called  the  outside  diameter  of  the  thread. 

1*1 


LONGITUDINAL   SECTION   OF   A    SCREW,   SHOWING   A   CROSS   SECTION 
OF    THE    THREADS 

Do,  outside  diameter;  Dr,  root  diameter;  d,  depth;  p,  pitch 

The  perpendicular  distance  from  the  top  of  the  groove  to 
the  bottom  is  called  the  depth  of  the  thread,  and  twice  this 
depth  is  called  the  double  depth  of  the  thread. 

The  bottom  of  the  groove  is  called  the  root  of  the  thread, 
and  the  diameter  of  the  screw  measured  at  the  bottom  of 
the  groove  is  called  the  root  diameter. 

The  distance  from  the  center  of  one  thread  to  the  center 
of  the  next  thread  is  called  the  pitch  of  the  screw  thread. 

The  advance  that  the  screw  makes  in  one  revolution  is 
called  the  lead  of  the  screw. 

35 


36 


SCREW  THREADS 


Multiple  Thread.  A  thread  formed  by  cutting  two  or  more 
uniform  spiral  grooves  around  a  bar  is  called  a  multiple  thread. 

A  multiple  thread  is  said  to  be  a  double  thread,  a  triple 
thread,  a  quadruple  -thread,  and  so  on,  according  as  the 
number  of  grooves  thus  cut  is  two,  three,  four,  and  so  on. 

The  pitch  and  the  lead  of  a  single  thread  are  equal,  but 
the  lead  of  a  double  thread  is  twice  the  pitch.  Similarly, 
in  a  triple  thread  the  lead  is  three  times  the  pitch,  in  a 
quadruple  thread  the  lead  is  four  times  the  pitch,  and  so  on. 


MULTIPLE    THREADS 

The  left-hand  figure  shows  a  double  thread  and  a  single  thread  of  the  same  lead, 

and  the  right-hand  figure  shows  a  triple  thread  and  a  single  thread  of  the  same 

lead.   In  each  figure  I  shows  the  lead  of  each  thread,  p  shows  the  pitch,  and  Dr 

shows  the  root  diameter 

Multiple  threads  are  often  used  where  machine  parts  are 
to  be  moved  by  the  action  of  a  screw.  If,  for  example,  a 
feed  screw  is  desired  which  shall  move  the  table  of  a  machine 
\"  per  revolution,  it  would  require  a  screw  of  large  diameter 
in  order  to  have  a  |-inch  pitch  and  fulfill  all  the  requisites 
for  strength  of  thread.  To  keep  the  diameter  of  the  screw 
to  an  appropriate  size  a  double  thread  of  ^-inch  pitch  might 
be  used,  and  the  desired  lead  of  \"  would  be  thus  obtained. 

Hand  of  a  Screw.  A  screw  that  advances  when  turned 
clockwise  is  called  a  right-hand  screw. 

A  screw  that  advances  when  turned  counterclockwise  is 
called  a  left-hand  screw. 

The  right-hand  screw  is  the  common  form,  and,  unless  otherwise 
stated,  a  right-hand  thread  is  always  understood  in  speaking  of  a  screw. 


MULTIPLE  THREAD  37 

Exercises.    Threads 

1.  A  screw  has  18  single  threads  to  the  inch.    Find  the 
pitch  and  the  lead  of  the  screw. 

2.  In  the  single-thread  screw  shown  on  page  35  suppose 
that  the  outside  diameter  is  lyg"  and  that  the  depth  of  the 
thread  is  -f^ff.    Find  the  root  diameter. 

It  is  evident  from  the  figure  that  the  root  diameter  is  equal  to  the 
outside  diameter  minus  the  double  depth  of  the  thread. 

3.  In  the  double-thread  screw  shown  on  page  36  suppose 
that  there   are   16   threads  to  the  inch ;   that  is,    8   double 
threads.    Find  the  pitch  and  the  lead  of  the  screw. 

4.  If  the  root  diameter  of  a  double-thread  screw  is  1-|" 
and  the  depth  of  the  thread  is  ^",  find  the  outside  diam- 
eter of  the  screw. 

5.  In  the  triple-thread  screw  shown  on  page  36  suppose 
that  the  pitch  is  -Jg-".    What  is  the  lead  of  the  screw  ? 

6.  The  cross  slide  on  a  lathe  is  moved  by  a  screw  that 
has  16  threads  per  inch,  and   one   revolution  of  the  screw 
moves  the  slide  \".    How  would  you  designate  the  thread? 

7.  A  lead  screw  has  a  triple  thread  with  18  threads  to  the 
inch.    Find  the  pitch  and  the  lead. 

8.  When  the  feed  screw  on  a  milling  machine  makes  one 
revolution  it  moves  the  table  ^".    If  the  screw  has  8  threads 
to  the  inch,  how  would  you  designate  the  thread  ? 

9.  How  many  revolutions  must  be  made  by  a  triple-thread 
feed  screw  of  i-inch  pitch  in  order  to  move  the  table  which 
it  controls  a  distance  of  11"  ? 

10.  When  the  screw  that  raises  the  knee  on  a  milling 
machine  makes  one  revolution  it  moves  the  knee  0.200".  If 
the  screw  has  5  threads  to  the  inch,  how  should  the  thread 
be  designated  ? 


SCREW  THREADS 


Sharp  V-Thread.    The  thread  of  which  the  cross  section  is 
an   equilateral    triangle   is    called    a   sharp  V-thread.     Since 
the  cross  section    of   this   thread 
is  an  equilateral  triangle,  all  the 
angles  are  60°.    The  depth  of  the 
thread  is  the  altitude  of  the  tri- 
angle, and  the  pitch  is  equal  to  the 
base,  as  is  evident  from  the  figure 
at  the  right. 

In  a  sharp  V-thread  the  pitch  and  *  Pitch  '>  d>  dePth 

the  depth  of  the  thread  are  found  by  the  following  formulas  : 


SHARP 


Pitch  = 


Depth  = 


1" 


number  of  threads  to  1" 

0.8660" 

number  of  threads  to  1" 


=  0.8660  x  pitch 


The  table  below  shows  the  standard  number  of  V-threads 
( T)  per  inch  for  screws  of  the  following  diameters  (^0)  : 


Da 

T 

Do 

T 

Do 

r 

D. 

7' 

r 

20 

\l" 

10 

ir 

5 

2|/x 

4 

A 

18 

7 

9 

1* 

42 

3 

H 

1 

16 

i$ 

9 

2 

41 

3^ 

'U 

A 

14 

1 

8 

2J 

4^ 

34 

8* 

i 

12 

H 

7 

2* 

4  1 

8| 

3i 

A 

12 

H 

7 

2f 

4^ 

8  4 

8i 

1 

11 

it 

6 

2£ 

4 

3f 

8} 

H 

11 

1} 

6 

2  5' 

4 

8j 

3 

1 

10 

if 

5 

2^ 

4 

4 

3 

Unless  otherwise  stated  the  diameter  of  a  screw,  as  D0  in  the  table 
above,  means  the  outside  diameter. 

Since  the  sharp  V-thread  offers  no  advantage  over  the  United  States 
Standard  thread  described  on  page  40  and  the  sharp  edges  are  easily 
injured,  its  manufacture  is  being  discontinued. 


SHAKP  V-THREADS  39 

Exercises.    Sharp  V -Threads 

1.  Find  the  depth  of  a  sharp  V-thread  with  a  pitch  of  J". 

In  the  following  problems  use  the  table  and  the  formulas  on  page  38. 
Carry  all  computations  involving  decimals  to  four  decimal  places,  but 
in  the  final  result  discard  the  fourth  place,  giving  the  dimension  correct 
to  the  nearest  0.001". 

2.  Find  the  double  depth  of  a  sharp  V-thread  with  9  threads 
to  the  inch ;  with  11  threads  to  the  inch. 

3.  Find  the  depth  of  a  sharp  V-thread  with  3J  threads  to 
the  inch;  with  4J  threads  to  the  inch. 

4.  Find  the  tap-drill  size  of  a  sharp  V-thread  tap  which 
has  an  outside  diameter  of  J". 

A  tap  is  a  tool  for  cutting  internal  threads,  and  the  tap-drill  size  is  the 
size  of  the  drill  for  boring  the  hole  to  be  tapped.  In  actual  practice,  to 
avoid  breaking  taps,  the  tap-drill  size  for  ordinary  work  is  generally 
taken  a  little  larger  than  the  root  diameter  of  the  thread  on  the  tap, 
but  in  all  problems  in  this  book  the  tap^drill  size  is  to  be  taken  as  equal 
to  the  root  diameter. 

5.  A  screw  having  5  sharp  V-threads  to  the  inch  has  a 
root  diameter  of  1.4036".    Find  the  outside  diameter. 

6.  Find  the  tap-drill  size  for  a  sharp  V-thread  tap  which 
has  an  outside  diameter  of  1|-". 

7.  If   the    root    diameter    of   a  sharp   V-thread   screw  is 
4.3072"  and  the  outside  diameter  is  5",  what  is  the  depth 
of  the  thread  and  the  number  of  threads  per  inch  ? 

Find  the  double  depth  of  a  sharp  V-thread,  given  that  the  pitch 
of  each  thread  is  as  follows : 

s.  Ty.       9.  ^".       10.  i".       11.  Ty.       12.  i". 

13.  Find  the  tap-drill  size  for  a  tap  If"  in  diameter  with 
6  double  sharp  V-threads  to  the  inch. 


40 


SCREW  THREADS 


United  States  Standard  Thread.    A  sharp  V-thread  flatted 
an  equal  amount  at  the  top  and  bottom,  as  here  shown,  is 
known  as  the  United  States  Stand- 
ard thread  and  is  commonly  des- 
ignated as  the  U.  S.  S.  thread. 

S.A.E.  Thread.  A  thread  of 
the  same  shape  as  the  U.  S.  S. 
thread,  but  which  differs  only  in 
the  number  of  threads  per  inch, 
bears  the  name  of  the  Society  of 
Automobile  Engineers  and  is  known  as  the  S.A.E.  thread. 

In  each  of  these  threads  the  following  formulas  are  used: 


UNITED    STATES    STANDARD 
THREAD 

d,  depth;  /,  flat;  p,  pitch 


Pitch  = 


Depth  = 


number  of  threads  to  1" 
0.6495" 


=  0. 6495  x  pitch 


number  of  threads  to  1" 
Flat  (top  and  bottom)  =  |  x  pitch 

The  table  below  shows  standard  U.S.S.  and  S.A.E.  threads: 


U.S.S.  THREAD 

S.A.E.  THREAD 

Do 

T 

D0 

T 

D0 

T 

D0 

T 

Do 

r 

r 

20 

V 

8 

2r 

4| 

I" 

28 

i" 
~s 

14 

A 

18 

11 

7 

2* 

4 

A 

24 

I 

14 

1 

16 

U 

7 

2| 

4 

3 
f 

24 

U 

12 

T7TT 

14 

If 

6 

3 

3^ 

T7* 

20 

1J 

12 

i 

13 

U 

6 

8J 

8j 

i 

20 

ii 

12 

T9* 

12 

If 

6i 

3^ 

?i 

T9* 

18 

U 

12 

i 

11 

*i 

5 

8| 

3 

1 

18 

.  . 

i 

10 

if 

5 

4 

3 

U 

16 

i 

9 

2 

n 

H 

27 
i 

i 

16 

In  this  table  D0  stands  for  the  diameter  of  the  screw  and  T  for  the 
number  of  threads  per  inch. 


U.S.S.  AND  S.A.E.  THREADS  41 

Exercises.    U.S.S.  and  S.A.E.  Threads 

1.  Find  from  the  formula  on   page  40   the  depth   of  a 
U.S.S.  thread  with  8  threads  to  the  inch. 

The  U.  S.  S.  thread  was  devised  by  William  Sellers  in  1869.  It  is  also 
known  as  the  Sellers  thread  and  as  the  Franklin  Institute  thread. 

2.  Find  the  tap-drill  size  for  a  l|-inch  S.A.E.  thread. 

3.  Find  the  double  depth  of  a  U.S.S.  thread  of  Jy-inch 
pitch;  of  y1^ -inch  pitch. 

4.  If  the   root  diameter  of  a  U.S.S.  thread  is  4.2551" 
and  there  are  2|  threads  to  the  inch,  what  is  the  outside 
diameter  of  the  screw? 

5.  What  is  the  root  diameter  of  an  S.A.E.  thread  ^"  in 
diameter?  |"  in  diameter? 

6.  Find  the  tap-drill  size  of  a  31-inch  U.S.S.  thread. 

7.  If  the  root  diameter  of  an  S.A.E.  thread  of  -J^-inch 
pitch  is  0.9072",  what  is  the  outside  diameter? 

Find  the  width  of  the  point  of  a  thread  tool  for  cutting  each 
of  the  following  numbers  of  U.  8.  S.  threads  to  the  inch  : 

8.  3.        9.  4|,        10.  7.        11.  13.        12.  18.        13.  20. 

The  width  of  the  point  of  the  cutting  tool  is  the  same  as  the  width 
of  the  flat  at  the  bottom  of  the  groove. 

14.  If  the  root  diameter  of  a  U.S.S.  thread  is  1.7113"  and 
the  outside  diameter  is  2",  what  is  the  depth  of  the  thread  ? 

15.  Find  to  the  nearest  ^"  the  proper  size  of  drill  for 
boring  a  hole  which  is   to  be   tapped  with   a  full   11-inch 
U.S.S.  thread;  with  a  full  f-inch  S.A.E.  thread. 

A  full  thread  is  a  thread  cut  to  the  depth  given  by  the  formula.  For 
ordinary  work  a  tap  drill  which  will  give  about  75%  of  a  full  thread  is 
generally  used,  and  tables  of  tap  drills  often  list  the  drill  sizes  in  sixty- 
fourths  of  an  inch  which  will  give  such  a  thread. 


42 


SCREW  THREADS 


Whitworth  Standard  Thread.  The  thread  of  which  the  cross 
section  is  of  the  form  here  illustrated  is  called  the  Whitworth 
Standard  thread. 

This  thread  was  devised  by  Sir 
Joseph  Whitworth  in  1841  and,  with 
slight  modifications,  is  still  the  British 
standard. 


In  such  a  thread  the  pitch,  the 
depth,  and  the.  radius  are  found 
by  the  following  formulas : 

1" 


WHITWORTH    STANDARD 
THREAD 

(7,  depth;  p,  pitch;  r,  radius  of 
curvature 


Pitch    = 


Depth  = 


Radius  = 


number  of  threads  to  1" 

0.6403" 

number  of  threads  to  1" 

0.1373" 
number  of  threads  to  1" 


=  0.6403  x  pitch 


=  0.1373  x  pitch 


The  following  table  shows  the  standard  number  of  threads 
per  inch  for  Whitworth  threads  of  the  following  diameters: 


Do 

T 

DO 

T 

D9 

T 

A, 

r 

A" 

60 

A" 

12 

u" 

7 

111" 

U 

£ 

48 

f 

11 

1A 

7 

2 

*J 

i 

40 

H 

11 

if 

6 

2i 

*1 

A 

32 

3 
? 

10 

i* 

6 

2J 

4 

A 

24 

« 

10 

H 

6 

2i 

4 

& 

24 

I 

9 

1A 

6 

2| 

3^ 

i 

20 

if 

9 

if 

5 

3 

»i 

A 

18 

1 

8 

itt 

5 

»i 

3i 

1 

16 

1A 

8 

if 

5 

3^ 

31 

A 

14 

H 

7 

iH 

5 

3^ 

3 

* 

12 

iA 

7 

ii 

H 

4 

3 

In  this  table  D0  stands  for  the  diameter  of  the  screw  and  T  for 
the  number  of  threads  per  inch. 


WHITWOETH  STANDARD  THREADS  43 

Exercises.   Whit  worth  Threads 

1.  Find  the  depth  of  a  Whitworth  thread  of  ^-inch  pitch. 

2.  Find  the  double  depth  of  a  Whitworth  thread  with 
7  threads  to  the  inch. 

Find  the  radius  for  the  point  of  the  tool  for  cutting  a  Whit- 
worth thread  of  which  the  pitch  in  each  case  is  as  follows : 

3.  0.4".        4.    i".        5.    i".        6.    i".        7.  -s\"'        8.  &"• 

9.  Find  the  tap-drill  size  for  a  Whitworth  thread  of  which 
the  outside  diameter  is  4". 

o 

10.  If  the  root  diameter  of  a  Whitworth  thread  is  3.3231" 
and   there   are   3  threads  to  the   inch,  what  is  the  outside 
diameter  of  the  screw  ? 

11.  If  the  root  diameter  of  a  Whitworth  thread  is  5.2377" 
and  there  are  2J  threads  to  the  inch,  what  is  the   outside 
diameter  of  the  screw  ? 

12.  Find  the  root  diameter  of  a  Whitworth  thread  with  an 
outside  diameter  of  Jg". 

13.  If  the  outside  diameter  of  a  Whitworth  thread  is  41" 
and  the  root  diameter  is  4.0546",  what  is  the  pitch  ? 

14.  find  the  root  diameter  of  a  Whitworth  thread  with  an 
outside  diameter  of  3£". 

15.  If  the  diameter  of  a  Whitworth  screw  is  ^",  how  many 
threads  are  there  to  the  inch  ?  What  is  the  pitch  ?  the  depth 
of  thread  ?  the  root  diameter  ? 

16.  If  the  root  diameter  of  a  Whitworth  thread  with  8 
threads  to  the  inch  is  0.9024",  what  is  the  outside  diameter? 

17.  If  the  diameter  of  a  Whitworth  screw  is  3",  how  many 
threads  are  there  to  the  inch  ?  What  is  the  pitch  ?  the  depth 
of  thread  ?  the  root  diameter  ? 


44 


SCREW  THREADS 


Metric  Threads.  A  metric  thread  is  similar  in  shape  to  the 
U.  S.  S.  thread  shown  on  page  40,  but  the  measurements  are 
in  millimeters  (1  mm.=  0.0394"). 

Of  the  two  standard  metric  threads 
in  general  use,  the  first  is  the  Inter- 
national Standard  thread,  adopted  at 
Zurich  in  1898,  and  the  second  is  the 
French  Standard.  These  threads  dif- 
fer but  little,  as  will  be  seen  by  com- 
paring the  measurements  of  the  threads  in  the  table  below. 

The  following  formulas  are  used  with  these  two  threads: 

Depth  =  0.6495  x  pitch 

Flat  (top  and  bottom)  =  |  x  pitch 

The  following  table  shows  standard  pitches  for  both  Inter- 
national Standard  and  French  Standard  metric  threads: 


METRIC    THREAD 

d,  depth ;  /,  flat ;  p,  pitch 


INTERNATIONAL  STANDARD 

FRENCH  STANDARD 

Diameter  of  screw 

Pitch 

Diameter  of  screw 

Pitch 

mm. 

in. 

mm. 

mm. 

in. 

mm. 

3 

0.1181 

0.55 

3 

0.1181 

0.50 

4 

.1575 

0.70 

4 

.1575 

0.75 

5 

.1969 

0.85 

5 

.1969 

0.75 

6 

.2362 

1.00 

6 

.2362 

1.00 

7 

.2756 

1.00 

7 

.2756 

1.00 

8 

.3150 

1.25 

8 

.3150 

1.00 

9 

.3543 

1.25 

9 

.3543 

1.00 

10 

.3937 

1.50 

10 

.3937 

1.50 

11 

.4331 

1.50 

11 

.4331 

1.50^ 

12 

.4724 

1.75 

12 

.4724 

1.50* 

14 

.5512 

2.00 

14 

.5512 

2.00 

16 

.6299 

2.00 

16 

.6299 

2.00 

18 

.7087 

2.50 

18 

.7087 

2.50 

20 

.7874 

2.50 

20 

.7874 

2.50 

22 

.8661 

2.50 

22 

.8661 

2.50 

METRIC  THREADS  45 

Exercises.   Metric  Threads 

1.  Find  from  the  formula  on  page  44  the  depth  of  a 
French  Standard  thread  of  1-millimeter  pitch. 

The  introduction  of  French  and  Italian  automobiles  into  this  country 
and  our  recent  expansion  of  foreign  trade  render  computation  in  metric 
units  necessary  as  well  as  desirable. 

In  problems  dealing  with  metric  threads  give  the  final  results  to  the 
nearest  0.1  mm.,  unless  otherwise  specified. 

2.  Find  the  double  depth  of  a  French  Standard  thread  of 
3-millimeter  pitch. 

3.  Find  the  root  diameter  of  a  10-millimeter  International 
Standard  thread. 

4.  Find  the  pitch  of  an  International  Standard  thread  of 
which  the  outside  diameter  is  27  mm.  and  the  root  diameter 
is  23.10  mm. 

5.  Find  the  outside  diameter  of  a  French  Standard  thread 
of   which   the   root   diameter   is   48.86  mm.    and    the   pitch 
is  5.50  mm. 

6.  For  particularly  accurate  work  the  formula  for  the 
depth  of   an  International   Standard   thread   is  as  follows: 
depth  =  0.64952  X  pitch.     If  the  pitch  is  4.500  mm.,  find  the 
depth  to  the  nearest  0.01  mm.,  using  this  formula. 

7.  In  Ex.  6  if  the  depth  is  0.974  mm.,  what  is  the  pitch? 

8.  Find  the  width  of  the  flat  of  an  International  Standard 
thread  with  an  outside  diameter  of  11  mm. 

9.  If  the  root  diameter  of  a  French  Standard  thread  is 
23.10  mm.  and  the  pitch  is  3  mm.,  what  is  the  outside  diam- 
eter of  the  screw  ? 

10.  If  the  outside  diameter  of  a  French  Standard  thread 
is  36  mm.  and  the  root  diameter  is  30.80  mm.,  what  is  the 
pitch  of  the  thread? 


46 


SCREW  THREADS 


Square  Thread.  The  thread  of  which  the  cross  section  is 
a  square,  as  shown  in  the  figure,  is  called  a  square  thread. 

In  theory  the  cross  sections  of  a 
square  thread  and  of  the  space  between 
two  successive  threads  are  both  squares. 
In  practice  the  groove  is  slightly  wider 
and  deeper  than  the  width  of  the  thread. 
For  our  present  purposes  we  shall  take  SQUARE  THREAD 

the  theoretical  measurements.  a,  depth;  /,  flat;  p,  pitch 

In  such  a  thread  the  pitch,  the 
depth,  and  the  flat  are  found  from  the  following  formulas: 

1" 

"  number  of  threads  to  1" 
Depth  =  flat  =  I  x  pitch 

Square-Thread  Tool.  The  tool  used  in  cutting  a  square 
thread  is  shaped  like  a  parting  tool,  or  cutting-off  tool,  except 
that  it  has  a  side  clearance,  or  ( / 

rake,  which  varies  for  the  pitch 
and  diameter  of  the  thread.    The 
distance    that    the    tool    travels 
around  the  work  is  the  root  cir- 
cumference  of   the   thread ;    the 
distance  that  it  travels  sideways 
is  the  pitch.   Letting  the  pitch  p  and  the  root  circumference  Cr 
be  two  sides  of  a  right  triangle,  as  in 
this  figure,  a  is  the  angle  for  grind- 
ing the  side  clearance  of  the  tool  and 
is  found  by  the  use  of  tangents. 

For  example,  find  the  angle  of  clearance  of  the  tool  for  cutting  a 
square  thread  of  ^-inch  pitch  on  a  shaft  2"  in  diameter. 

»  0.25  0.25 

tan  a  =  —  = = =  0.0455. 

Cr      3.1416  x  (2  -  0.25)      5.4978 

Therefore  a  =  2°  36'. 


^ 


SQUARE-THREAD    TOOL 
a,  angle  for  grinding  the  tool 


SQUARE  THREADS  47 

Exercises.    Square  Threads 

1.  From  the  formulas   on  page  46    find    the  pitch,   the 
depth,  the  double  depth,  and  the  flat  of  a  square  thread  with 
8  threads  to  the  inch. 

2.  If  the  outside  diameter  of  a  screw  with  square  threads 
is  2|",  and  there  are  8  threads  to  the  inch,  what  is  the  root 
diameter  of  the  screw  ? 

3.  At  what  angle  should  the  tool  be  ground  for  cutting 
the  thread  referred  to  in  Ex.  2? 

4.  If  a  double  square  thread  has  a  diameter  of  iy  and 
a  lead  of  J",  what  is  the  root  diameter? 

5.  In  cutting  a  square  thread  with  2  threads  to  the  inch 
on  a  shaft   41"  in   diameter,  what   angle  of  side  clearance 
should  be'  allowed  for  the  tool? 

6.  If  the  double  depth  of  a  square  thread  is  0.0417",  what 
is  the  number  of  threads  to  the  inch  ? 

7.  Find  the  angle  for  grinding  the  side  clearance  of  the 
tool  used   in   cutting   a   square   thread   having   an    outside 
diameter  of  1|"  and  a  pitch  of  -^g". 

8.  A  triple  square-thread  screw  of  outside  diameter  1|" 
has  a  lead  of  \".    Find  the  root  diameter  of  the  thread. 

9.  Find  the  angle  for  grinding  the  side  clearance  of  the 
tool  for  cutting  the  thread  in  Ex.  8. 

10.  Find  the  outside  diameter  of  a  square  thread  which 
has  a  root  diameter  of  3T5g-"  and  a  pitch  of  -J". 

11.  Find  the  angle  for  grinding  the  tool  used  in  cutting 
the  thread  described  in  Ex.  10.    What  would  be  the  angle 
if  the  thread  were  a  double  thread  of  the  same  lead? 

12.  If  the  root  circumference  of  a  square  thread  is  2.3562" 
and  the  pitch  is  J",  what  is  the  outside  diameter? 


48  SCKEW  THREADS 

Acme  Thread.  The  thread  illustrated  in  the  figure  below 
is  called  an  Acme  thread.  The  nature  of  this  thread  will  be 
understood  from  the  illustration.  Nut 

Owing  to  the  ease  of  cutting  and  to 
the  greater  strength  secured,  the  Acme 
thread  has  to  a  large  extent  replaced 
the  square  thread. 

From  the  formulas  given  below  it 
will  be  seen  that  the  depth  of  the  Acme  ACME  THREAD 

screw  thread  differs  by  0.0100"  from 
that  of  the  square  thread.  d'  depth;  p'  pit( 

The  following  formulas  are  used  in  computing  measure- 
ments of  Acme  screw  threads  and  tap  threads: 


number  of  threads  to  1" 
Depth  for  screws  =|  x  pitch  +  0.0100" 
Depth  for  taps  =  |  x  pitch  +  0.0200" 

Width  of  flat  at  root  for  screws  =  0.3707  X  pitch  —  0.0052" 
Width  of  flat  at  root  for  taps  =  0.3707  x  pitch  -  0.0052" 
Width  of  flat  at  top  for  screws  =  0.3707  x  pitch 
Width  of  flat  at  top  for  taps  =  0.3707  x  pitch  —  0.0052" 
Width  of  space  at  top  for  screws  =  0.6293  x  pitch 
Width  of  space  at  top  for  taps  =  0.6293  x  pitch  +  0.0052" 
Diameter  of  tap  =  diameter  of  screw  +  0.0200" 
Root  diameter  for  screws  and  taps  =D0  —  (pitch  +  0.0200") 

The  symbol  D0  in  the  last  formula  stands  for  "  outside  diameter." 

It  is  evident  from  the  above  formulas  that  the  clearance 
at  the  bottom  of  an  Acme  thread  between  the  nut  and  the 
screw  is  obtained  by  making  the  outside  diameter  of  the 
nut  thread  0.0200"  larger  than  that  of  the  screw  thread. 


ACME  THREADS  49 

Exercises.    Acme  Threads 

1.  Find  from  the  formula  on  page  48  the  width  of  the 
flat  at  the  root  of  an  Acme  screw  thread  having  2J-  threads 
to  the  inch. 

2.  Find  the  depth  of  an  Acme  tap  thread  of  ^-inch  pitch. 

3.  Find  the  width  of  the  flat  at  the  top  of  an  Acme  screw 
thread  having  12  threads  to  the  inch. 

4.  Find  the  root  diameter  of  an  Acme  tap  thread  having 
4  threads  to  the  inch  and  an  outside  diameter  of  2J". 

5.  Find  the  outside  diameter  of  an  Acme  screw  thread 
having  a  root  diameter  of  3.4443"  and  3J  threads  to  the  inch. 

6.  A  boring  bar  with  a  diameter  of  3|"  is  threaded  with 
an  Acme  screw  thread  which  has  a  double  depth  of  0.2422". 
Find  the  pitch  of  the  thread. 

7.  In  Ex.  6  find  the  width  of  the  flat  at  the  root. 

8.  Find  the  width  of  the  flat  at  the  top  and  also  at  the 
root  of  an  Acme  screw  thread  of  |-inch  pitch. 

9.  Find  the  root  diameter  of  an  Acme  tap  thread  which 
has  an  outside  diameter  of  1-jV  and  7  threads  to  the  inch. 

10.  In  an  Acme  screw  thread  of  -j3g-inch  pitch  find  the 
width  of  the  flat  at  the  top,  the  width  of  the  flat  at  the  root, 
and  the  width  of  the  space  at  the  top. 

11.  If  the  outside  diameter  of  an  Acme  screw  thread  hav- 
ing 3  threads  to  the  inch  is  ij",  what  is  the  root  diameter  ? 
the  outside  diameter  of  the  corresponding  tap  thread  ? 

12.  If   the   root    diameter   of    an   Acme    screw  thread   is 
3.2982"  and  the  outside  diameter  is  4J",  what  is  the  pitch 
of  the  thread  ? 

13.  Find  the  outside  diameter  of  an  Acme  screw  thread 
with  12  threads  to  the  inch  and  a  root  diameter  of  1.1467". 


50  SCREW  THREADS 

Briggs  Pipe  Threads.  A  Briggs  pipe  thread  differs  from  the 
threads  so  far  considered  in  that  for  a  certain  distance,  A~B 
in  figure  III  below,  the  outside  of  the  pipe  is  tapered.  The 
threads  P,  which  are  on  the  taper,  are  perfect  threads,  the 


in 

BRIGGS   PIPE   THREADS 

Figure  I  shows  a  cross  section  of  a  perfect  standard  thread  as  devised  by  Robert 

Briggs,  a  British  engineer,  in  1882 ;  figure  II  shows  a  similar  cross  section  of  the 

modified  Briggs  thread ;  figure  III  shows  a  longitudinal  section  of  the  complete 

thread,  the  letters  in  this  figure  being  explained  in  the  text 

next  two  threads  /,  not  being  on  the  taper,  are  imperfect  on 
top,  and  the  next  four  threads  I'  are  imperfect  at  the  bottom 
as  well,  owing  to  the  chamfer  of  the  threading  die. 

The  nominal  inside  diameter  (N.I. D.)  is  the  size  under 
which  the  pipe  is  listed  and  differs  considerably  from  the 
actual  inside  diameter  (A.I.D.),  which  is  the  exact  measure- 
ment, as  is  seen  in  the  table  on  page  51.  The  actual  outside 
diameter  (A.  O.D.)  is  measured  at  the  large  end  of  the  taper, 
which  has  a  T.P.F.  of  |"  between  points  A  and  B. 

The  following  formulas  are  used  with  Briggs  threads  and 
show  the  differences  between  the  standard  and  modified  forms : 

1" 
Pitch 


number  of  threads  to  1" 
Depth  (standard)  =  0.8  x  pitch 
Depth  (modified)  =  0.833  x  pitch 
Flat  (modified)  =  i  x  pitch 

The  modified  form  of  Briggs  thread  is  more  easily  cut  and  hence 
has  come  into  more  general  use. 


BRIGGS  PIPE  THREADS 


51 


Table  of  Briggs  Threads.   The  table  below  shows  the  stand- 
ard measurements  and  the  number  of  threads  (T7)  per  inch: 


N.I.D. 

A.I.D. 

A.O.D. 

T 

N.I.D. 

A.I.D. 

A.O.D. 

T 

i" 

0.270" 

0.405" 

27 

2" 

2.067" 

2.375" 

Hi 

.364 

.540 

18 

21 

2.468 

2.875 

8 

I 

.494 

.675 

18 

3 

3.067 

3.500 

8 

.623 

.840 

14 

31 

3.548 

4.000 

8 

a 

.824 

1.050 

14 

4 

4.026 

4.500 

8 

1 

1.048 

1.315 

111 

4| 

4.508 

5.000 

8 

1J 

1.380 

1.660 

Hi 

5 

5.045 

5.563 

8 

u 

1.610 

1.900 

Hi 

6 

6.065 

6.625 

8 

Exercises.    Briggs  Pipe  Threads 

1.  Find  the  depth  of  a  modified  Briggs  pipe  thread  hav- 
ing 27  threads  to  the  inch. 

2.  Find    the    double    depth   of   a   modified   Briggs   pipe 
thread  having  111  threads  to  the  inch. 

3.  Find  the  width  of  the  flat  of  a  modified  Briggs  pipe 
thread  with  8  threads  to  the  inch. 

Find  the  root  diameter  at  the  large  end  of  the  taper  of  a  Briggs 
Standard  thread  for  each  of  the  following  sizes  of  pipes  (N.  /.  Z>.)  : 

4.  \".         5.  \".         6.  1".         7.  II".        8.  3".         9.  4". 

10.  In  the  Briggs  Standard  thread  the  length  of  the  per- 

,    0.8  D  +  4.8 


feet  thread,  P  in  figure  III  on  page  50,  is 


N 


,  where 


D  is  the  A.O.D.  and  N  is  the  number  of  threads  per  inch. 
Find  the  length  of  the  perfect  thread  for  a  |-inch  pipe. 

As  in  Ex.  10,  find  the  length  of  the  perfect  thread  in  each 
case  for  pipes  of  the  following  sizes  : 

11.  £" .       12.   V.       13.   1J".       14.   1J".      15.   2".      16.   31". 


52 


SCREW  THREADS 


Measuring  Threads.  In  addition  to  the  outside  diameter 
and  the  root  diameter  of  a  screw,  there  is  also  the  pitch 
diameter,  or  the  angle  diameter,  which  is  used  in  measuring 
threads.  The  pitch  diameter  is  theoretically  the  outside 
diameter  minus  the  single  depth  of  the'  thread;  that  is,  it 
includes  half  the  depth  of  the  thread  on  opposite  sides  of 
the  screw. 

The  best  means  of  measuring  the  pitch 
diameter  of  a  thread  is  by  the  use  of  a 
special  thread  micrometer.  In  such  an 
instrument  the  anvil  is  V-shaped  so  as  to 
fit  over  the  thread,  and  the  screw,  pointed 
to  60°,  is  slightly  rounded  so  as  to  enter 
the  groove. 

In  practice,  however,  the  pitch  diameter 
is  not  actually  measured,  but  is  tested  by 
the  use  of  the  ordinary  micrometer  and  of    TESTING  THE  PITCH 
three  wires  of  the  same  diameter  placed  as      DIAMETER  OF  A 

TJ»  S«  S»   THREAD 

here  shown.   The  wires  used  are  hardened, 

M,  M,  micrometer  jaws; 

tempered,  accurately  ground,  and  lapped         w,  w,  w,  wires 
to  size.    One  wire  is  placed  in  the  angle 
of  the  thread,  and  the  other  two  wires  are  placed  in  adjacent 
angles  on  the  opposite  side.    The  micrometer  is  placed  over 
the  wires,  and  the  reading  is  taken. 

By  trigonometry  it  can  be  shown  that  if  M  is  the  measure- 
ment over  the  wires,  D0  the  outside  diameter  of  the  screw, 
p  the  pitch,  C  a  certain  constant,  and  Dw  the  diameter  of 
the  wires,  the  following  formula  is  true  : 


The  pitch  diameter  of  the  screw  is  correct  (that  is,  it  is 
equal  to  the  theoretical  pitch  diameter)  when  the  micrometer 
reading  over  the  wires  is  equal  to  the  result  from  the  formula. 


MEASURING  THEEADS 


53 


Illustrative  Problems.  1.  Find  the  correct  measurements 
over  wires  of  a  £-inch  U.  S.  S.  thread,  the  diameter  of  the 
measuring  wires  being  0.05773"  and  the  constant,  which  is 
used  for  all  U.  S.  S.  threads,  being  1.5155. 

We  first  refer  to  the  table  of  U.  S.  S.  threads  on  page  40,  and  find  that 
when  the  outside  diameter  of  the  thread  is  £"  there  are  10  threads  to 
the  inch.  The  pitch,  therefore,  is  ^". 

Using  the  formula  given  on  page  52,  we  find  M,  the  measurement 
over  the  wires,  as  follows  : 


M=  D0-pC+  3  Dw  =  0.75  - 


1.5155 
10 


+  3  x  0.05773  =  0.77164 


Since  a  vernier  micrometer  gives  readings  only  to  0.0001"  we  give 
the  above  result  as  0.7716".  If  the  reading  of  the  caliper  agrees  with 
this  result,  the  pitch  diameter  is  correct. 

2.  Find  the  correct  micrometer  reading  over  wires  for  a 
T5g-inch  sharp  V-thread,  the  diameter  of 
the  wires  being  0.035"  and  the  constant 
for  sharp  V-threads  being  1.732. 

For  sharp  V-threads  we  find  on  page  38 
that  when  D  =  T5^"  the  number  of  threads  is 
18,  and  hence  the  pitch  is  TV-  We  therefore 

have 


+  3  x  0.035 


=  0.3125  - 
=  0.3213. 


TESTING        THE        PITCH 

The  correct  reading  for  a  ^-inch  sharp  DIAMETER  OF  A  SHARP 
V-thread  is  therefore  0.3213".  V-THREAD 

Use  of  the  Three-Wire  Method.  If,  in  cutting  the  sharp 
V-thread  described  above,  it  is  desired  to  test  the  depth  of 
cut,  the  wires  are  placed  as  shown  in  the  figure  and  the  mi- 
crometer reading  is  taken  over  them.  If,  for  example,  this 
reading  is  0.01"  more  than  0.3213",  the  thread  must  be  cut 
0.01"  deeper  to  be  correct ;  if  the  reading  is  less  than  0.3213", 
the  thread  has  been  cut  too  deep. 


54 


SCKEW  THREADS 


Exercises.    Measuring  Threads 

1.  Find    the   correct   reading    over    wires    for    a    2 1 -inch 
Whitworth  thread,  the  wires  being  0.150"  in  diameter  and 
the  constant  for  this  thread  being  1.6008. 

The  formula  for  the  correct  reading  of  a 
Whitworth  thread  differs  from  the  one  given  on 
page  52.  It  is  as  follows  : 

M  =  D0-pC  +  3.1657  Dw 

For  the  pitch  see  the  table  on  page  42. 

2.  Find  the  correct  reading  over  wires 
for  a  1^-inch  U.S.S.  thread,  using  wires 
0.090"  in  diameter. 

3.  Find  the  correct  reading  over  wires     TESTING  THE   PITCH 
for   a   sharp   V-thread   If"   in    diameter,     DIAMKTEBOFA  WHIT- 

r  •         ••-  WORTH  THREAD 

using  wires  0.150"  in  diameter. 

4.  Find  the  correct  reading  over  wires  for  a  U.S.S.  thread 
3"  in  diameter,  using  wires  0.200"  in  diameter. 

5.  Copy  the  following  table  and  in  the  proper  space  insert 
the  correct  measurement  over  wires  for  each  thread : 


U.  S.  S.  THREAD 

WHITWORTH  THREAD 

Outside 
diameter 

Diameter 
of  wires 

Measure- 
ment over 
wires 

Outside 
diameter 

Diameter 
of  wires 

Measure- 
ment over 
wires 

IM 

2 

0.050" 

_7_" 
16 

0.04026" 

1 

0.06415 

1  3 
*8 

0.09394 

If 

0.10497 

2 

0.12526 

2i 

0.12830 

8i 

0.17344 

H 

0.17623 

3| 

0.18789 

REVIEW  EXERCISES  55 

Exercises.    Review 

1.  Find   the  lead   of   a   double-thread    screw   which   has 
12  threads  to  the  inch;  which  has  20  threads  to  the  inch. 

2.  Find  the  root  diameter  of  an  S.A.E.  thread  that  is 
11"  in  diameter. 

3.  Find   the   double   depth   of   a   Briggs    Standard   pipe 
thread  for  a  4 1 -inch  pipe. 

4.  In  an  Acme  screw  thread  with  a  pitch  of  Ty  find  the 
width  of  the  flat  at  the  top,  the  width  of  the  flat  at  the  root, 
and  the  width  of  the  space  at  the  top. 

5.  What  is  the  pitch  of  a  square  thread  when  the  double 
depth  is  0.2222"?  How  many  threads  are  there  to  I"  ? 

6.  If   the   root   diameter   of   an    International    Standard 
thread  is  5.70  mm.  and  the  pitch  is  1  mm.,  what  is  the  outside 
diameter  ?  the  width  of  the  flat  ? 

7.  At  what  angle  should  the  thread  tool  be  ground  to 
cut  a  square  thread  with  6  threads  to  the  inch  and  an  out- 
side diameter  of  1^"? 

8.  Find  the  tap-drill  size  of  a  l|-inch  U.S.S.  tap. 

9.  Find  the  root  diameter  of  an  Acme  tap  thread  2-|"  in 
outside  diameter  with  5  threads  to  the  inch. 

10.  Find  the  correct  reading  over  wires  for  a  sharp  V-thread 
11"  in  diameter,  the  diameter  of  the  wires  being  0.070". 

11.  The  root  diameter  of  a  U.S.S.  thread  with  3J  threads 
to  the  inch  is  2.6288".    Find  the  outside  diameter. 

12.  The  root  diameter  of  a  Whitworth  Standard  thread 
is  4.5343"  and  the  outside  diameter  is  5".    Find  the  pitch 
of  the  thread. 

13.  If  the  outside  diameter  of  an  Acme  screw  thread  hav- 
ing 11  threads  to  the  inch  is  31",  what  is  the  root  diameter  ? 


56 


SCEEW  THREADS 


Screw-Thread  Cutting.  The  cutting  of  a  thread  in  a  lathe 
is  a  mechanical  operation  in  which  the  selection  of  the 
proper  change  gears 
is  a  matter  of  greatest 
importance. 

In  order  to  cut  the 
required  number  of 
threads  per  inch,  the 
number  of  revolutions 
of  the  work  must  be 
exactly  equal  to  the 
number  of  threads  to 
be  cut  during  the  time 
that  the  carriage  carry- 
ing the  tool  rest  moves  LATHE  HEADSTOCK  SHOWING  SIMPLE  GEAR- 
exactly  1"  along  the  bed  ING  FOR  CUTTING  THREA»S 

of  the  lathe.     Change    A> lathe  sPindle;  B> lead  screw;  c,  stud  shaft; 

E,  intermediate  gear 

gears   are  used  so   as 

to  give  the  proper  number  of  revolutions  to  the  lead  screw, 
which  moves  the  carriage,  gears  of  different  sizes  being  placed 
on  the  stud  shaft  and  on  the  lead  screw  according  to  the 
size  of  thread  to  be  cut.  For  fine  threads  the  carriage  moves 
slowly,  but  for  coarse  threads  it  moves  rapidly  along  the  work. 
A  simple-geared  lathe  has  only  one  change  of  speed  be- 
tween the  stud  shaft  and  the  lead  screw. 

Since  the  gear  on  the  lathe  spindle  has  the  same  number  of  teeth  as 
the  driven  gear  on  the  stud  spindle,  the  two  spindles  have  the  same  speed. 
The  intermediate  gear  connecting  the  driving  gear  on  the  stud  and 
the  gear  on  the  lead  screw  may  be  of  any  size  since  it  has  no  effect  on 
the  rate  of  turning  of  the  lead  screw. 

The  lead  of  the  lathe  can  be  found  by  counting  the  threads  per  inch 
on  the  lead  screw,  or  by  placing  gears  of  the  same  number  of  teeth  on 
the  stud  shaft  and  on  the  lead  screw  and  then  counting  the  number  of 
revolutions  of  the  lead  screw  while  the  carriage  moves  1". 


SCKEW-THKEAD  CUTTING  57 

Illustrative  Problem.  If  the  lead  screw  of  a  lathe  has 
6  threads  per  inch  and  it  is  desired  to  cut  12  threads  per 
inch  on  a  screw,  what  change  gears  are  required  ? 

In  this  case,  since  the  lead  screw  will  revolve  6  times  while  the  car- 
riage advances  1",  and  the  work  must  revolve  12  times  during  the  same 
period,  change  gears  with  a  ratio  of  6  to  12,  or  of  1  to  2,  should  be 
placed  on  the  stud  shaft  and  the  lead  screw.  That  is,  expressed  as  a 
general  proportion,  which  may  be  used  in  any  problem  requiring  the 
use  of  simple  gearing,  we  have  the  following : 

Number  of  teeth  on  outside  stud  gear 
Number  of  teeth  on  lead-screw  gear 

_  number  of  threads  per  inch  on  lead  screw 
number  of  threads  per  inch  on  work 

Each  term  of  the  ratio  of  the  gears  is  multiplied  by  a  number  such 
that  the  result  is  a  ratio  whose  terms  correspond  to  the  numbers  of 
teeth  on  two  of  the  gears  with  which  the  lathe  is  equipped. 

4x6       24 

In  this  case  we  have          = 

4  x  12      48 

That  is,  a  gear  with  24  teeth  should  be  placed  on  the  end  of  the  stud 
shaft,  and  a  gear  with  48  teeth  on  the  lead  screw.  An  intermediate  gear 
of  the  proper  size  is  used  to  connect  the  two  gears. 

Variation  of  Lathes.  The  number  of  threads  per  inch  on 
the  lead  screw  varies  on  different  makes  of  lathes,  and  this 
makes  it  necessary  to  have  different  sets  of  change  gears. 
In  our  problems  we  shall  use  the  following  sets  of  gears : 

For  a  lead  screw  with  5  threads  per  inch :  25,  25,  30,  35, 
40,  45,  50,  55,  60,  65,  69,  70,  80,  90,  100,  110,  120. 

FoY  a  lead  screw  with  6  threads  per  inch :  24,  24,  32,  40, 
44,  48,  52,  56,  60,  64,  72,  110. 

For  a  lead  screw  with  8  threads  per  inch :  24,  24,  28,  32, 
36,  40,  44,  48,  52,  56,  64,  69,  72,  80,  96. 

In  the  exercises  use  only  gears  which  are  shown  above  as  belonging 
to  the  set  for  the  lead  screw  specified  in  each  problem. 


58  SCREW  THREADS 

Exercises.   Screw-Thread  Cutting 

1.  What  change  gears  are  required  on  a  lathe  with  a  lead 
screw  of  1-inch  pitch  in  cutting  a  screw  thread  of  -Jg-inch 
pitch  ?   in  cutting  a  thread  with  a  pitch  of  0.125"  ? 

Find  the  change  gears  required  on  a  lathe  with  a  lead  screw 
of  A-inch  pitch  for  cutting  each  of  the  following  numbers  of 
threads  per  inch: 

2.  12.  3.   7.  4.  13.  5.  18.  6.  111. 

Find  the  change  gears  required  on  a  lathe  having  a  lead  screw 
with  8  threads  per  inch  for  cutting  each  of  the  following  num- 
bers of  threads  per  inch : 

7.  4.  9.  9.  11.  12.  13.  18.  15.  10. 

8.  5.  10.  16.  12.  7.  14.  6.  16.  20. 

17.  What  is  the  pitch  of  the  thread  that  is  cut  on  a  lathe 
with  a  lead  screw  of  J-inch  pitch  when  there  is  a  25-tooth 
gear  on  the  stud  and  a  40-tooth  gear  on  the  lead  screw  ? 

Find  the  change  gears  required  on  a  lathe  having  a  lead  screw 
with  6  threads  per  inch  for  cutting  each  of  the  following  num- 
bers of  threads  per  inch  : 

18.  6.  19.  10.  20.  13.  21.   18.  22.  271. 

23.  Find  the  largest  number  of  threads  per  inch  that  can 
be  cut  with  simple  gearing  on  a  lathe  which  has  a  lead  screw 
of  ^-inch  pitch ;  of  1-inch  pitch ;  of  ^-inch  pitch. 

24.  Find  the  smallest  number  of  threads  per  inch  that  can 
be  cut  with  simple  gearing  on  a  lathe  which  has  a  lead  screw 
of  ^-inch  pitch ;  of  1-inch  pitch ;  of  ^-inch  pitch. 

25.  Find  the  change  gears  necessary  to  cut  a  thread  with 
a  lead  of  f  "  if  the  lead  screw  of  the  lathe  has  5  threads  per 
inch ;  if  the  lead  screw  has  8  threads  per  inch. 


COMPOUND  GEAKING 


59 


Compound  Gearing.  A  compound  gearing  consists  of  two 
gears  keyed  together  to  revolve  at  the  same  rate  and  held 
in  position  by  a  bracket.  The  gears  are  introduced  between 
the  gear  on  the  stud  and 
that  on  the  lead  screw 
and  are  usually  in  the 
ratio  of  2  to  1,  but  the 
general  principle  of  oper- 
ation is  the  same  with 
gears  of  any  other  ratio. 

Since  the  cutting 
of  a  very  small  or  a 
very  large  number  of 
threads  per  inch  with 
simple  gearing  would 
require  the  use  of  gears 
with  a  very  large  num- 
ber of  teeth,  it  is  neces- 
sary to  use  compound 
gearing  on  a  lathe. 

Illustrative  Problem.  Find  the  change  gears  required  on 
a  lathe  with  a  lead  screw  of  -^-inch  pitch  in  cutting  a  screw 
thread  with  30  threads  per  inch. 

Solving  as  for  simple  gearing,  the  ratio  of  the  gears  is  found  to  be 
3%-,  and  since  a  25-T  gear  is  the  smallest  one  that  goes  with  a  lead 
screw  of  ^-inch  pitch,  we  shall  need  a  gear  with  150  teeth  if  we  use  only 
simple  gearing.  To  avoid  this  we  split  the  ratio  ^o  into  factors,  thus : 

5  _  1  _  1  (gear  on  stud)  1  (driving  gear  on  compound) 

30      6      3  (gear  on  lead  screw)      2  (driven  gear  on  compound) 

We  therefore  see  that  the  gears  on  the  stud  and  the  lead  screw  must 
have  the  ratio  of  1  to  3,  such  as  30  T  to  90  T,  while  the  gears  on  the  com- 
pound must  have  the  ratio  of  1  to  2,  such  as  25  T  to  50  T,  the  smaller 
gear  being  the  driving  gear  and  the  larger  gea'r  being  the  driven  gear. 


LATHE  HEADSTOCK  SHOWING  COMPOUND 
GEARING  FOR  CUTTING  THREADS 

A,  lathe  spindle;  B,  lead  screw;   C,  stud  shaft; 

D,  first  gear  on  compound,  driven;  E,  second 

gear  on  compound,  driving 


60  SCREW  THREADS 

Exercises.    Screw-Thread  Cutting 

1.  Find  the  change  gears  required  on  a  lathe  with  a  lead 
screw  of  i-inch  pitch  in  cutting  25  threads  to  the  inch. 

This  example  shows  that  the  gears  on  the  compound  are  not  neces- 
sarily in  the  ratio  of  2  to  1,  the  principle  being  the  same  for  any  ratio. 

Find  the  change  gears  required  on  a  lathe  with  a  lead  screw 
of  —-inch  pitch  in  cutting  a  thread  with  each  of  the  following 
numbers  of  threads  to  the  inch : 

2.  28.  3.   7|.  4.  3|.  5.  32.  6.  42. 

Similarly,  for  a  lead  screw  of  ^-inch  pitch,  the  numbers  being : 
7.  1.  8.  11.  9.  22.          10.  36.  11.  28. 

Similarly,  for  a  lead  screw  of  ^-inch  pitch,  the  numbers  being : 
12.  2.  13.   23.  14.  T5r  15.  f.  16.  1^. 

Find  the  change  gears  required  on  a  lathe  having  a  lead 
screw  with  5  threads  to  the  inch  in  cutting  a  thread  with  each 
of  the  following  leads  : 

17.  Ty.       is.  f".       19.  f".       20.  f.       21.  Ty. 

22.  Find  the  change  gears  required  on  a  lathe  having  a 
lead  screw  with  6  threads  to  the  inch  in  cutting  a  metric 
screw  with  a  lead  of  2  mm. 

First  find  the  number  of  threads  to  the  inch  (1"  =  25.4  mm.)  corre- 
sponding to  the  given  lead  in  millimeters.  It  will  be  found  that  a  gear 
with  127  teeth  is  required  to  cut  a  metric  thread  on  a  lathe  which  has 
an  English  lead  screw.  In  Exs.  22  and  23  assume  that  the  lathe  is 
equipped  with  such  a  gear  in  addition  to  those  listed  on  page  57. 

23.  On  a  lathe  having  a  lead  screw  with  6  threads  to  the 
inch,  find  the  change  gears  required  for  cutting   a  metric 
thread  with  a  lead  of  3  mm.;  with  a  lead  of  3.5  mm. 


CHAPTER  V 

INDEXING  AND  SPIRAL  CUTTING 

Indexing.  The  process  of  dividing  a  circumference  into 
equal  parts  is  called  indexing. 

Indexing  is  done  by  the  aid  of  a  milling-machine  fixture 
known  as  the  dividing  head,  spiral  head,  or  index  head,  and 
which  is  described  on  page  63.  The  dividing  head  is  ordinarily 
employed  in  obtaining  equal  divisions  on  a  circumference,  but 
may  be  used  to  obtain  equal  divisions  of  various  kinds  on 
other  types  of  work. 

There  are  various  kinds  of  indexing,  —  direct,  simple, 
angular,  compound,  and  differential.  These  will  be  considered 
and  explained  in  the  following  pages. 

Among  the  cases  in  which  the  machinist  will  make  use  of 
indexing  are  the  following :  in  cutting  spur,  bevel,  and  spiral 
gears ;  in  cutting  worm  wheels ;  in  cutting  plate  and  cylin- 
dric  cams;  and  in  making  milling-machine  cutters,  counter 
bores,  reamers,  and  twist  drills. 

The  dividing-head  spindle  may  be  set  at  any  angle  from  about  5° 
below  the  horizontal  to  about  10°  beyond  the  vertical.  The  swiveling 
block  is  usually  graduated  about  its  circumference  in  fourths  of  a 
degree,  but  some  machines  have  a  vernier  that  reads  to  ^°. 

Work  is  held  in  a  dividing  head  in  a  manner  similar  to  that  used  on 
an  engine  lathe,  the  three  following  general  methods  being  used  :  (1)  by 
mounting  it  on  centers,  (2)  by  holding  it  on  an  arbor,  and  (3)  by  secur- 
ing it  in  a  chuck. 

When  the  work  is  held  on  an  arbor  or  mounted  on  centers  it  is 
necessary  to  use  the  dividing-head  tailstock,  which  has  an  adjustable 
center  capable  of  being  raised  or  lowered. 

61 


62 


INDEXING  AND  SPIRAL  CUTTING 


Index  Center.  The  index  center  shown  below  is  used  to 
obtain  exact  and  rapid  spacing  on  work  that  requires  only  a 
small  number  of  divisions  upon  the  periphery,  as  is  the  case 
when  milling  bolt  heads  and  fluting  reamers  and  taps.  It  is 
used  only  on  machines  that  are  not  fitted  with  a  dividing 
head,  since  the  dividing  head  will  do  all  that  an  index 
center  can  perform,  and  has  many  additional  advantages. 


INDEX    CENTER 
A,  spindle ;  B,  index  plate ;  C,  headstock ;  D,  footstock ;  E,  stop  pin 

The  index  center  consists  of  two  main  parts,  the  headstock 
C  and  the  footstock  D.  The  spindle  A  is  moved  freely  by 
the  index  plate  B,  which  is  fastened  to  it.  The  spindle  is 
locked  at  the  desired  division  by  means  of  the  stop  pin  E, 
which  is  attached  to  the  headstock.  There  is  an  even  number 
of  grooves,  usually  24,  36,  or  48,  on  the  circumference  of 
the  index  plate,  and  the  number  of  divisions  which  can  be 
made  on  the  work  is  limited  to  the  number  of  grooves  and 
to  factors  of  this  number. 

Direct  Indexing.  It  is  seen  from  the  above  description  that 
the  index  plate  gives  direct  motion  to  the  spindle,  and  the  use 
of  the  instrument  in  this  way  is  called  direct  indexing. 


DIVIDING  HEAD 


63 


Dividing  Head.  Instead  of  connecting  the  index  plate  B 
directly  to  the  spindle  A,  as  in  the  index  center,  the  index 
crank  F  of  the  dividing  head  here  shown  is  connected  to 


DIVIDING    HEAD 

A,  spindle ;  B,  index  plate ;  H,  index  pin 

shaft  E,  on  which  is  cut  a  single-thread  worm  D.  The  worm 
meshes  with  the  40-tooth  worm  wheel  C  on  the  spindle,  thus 
transmitting  the  motion  from 
the  crank. 

The  index  plate  B  is  not 
fastened  to  shaft  E  and  can 
be  held  stationary  by  the  stop 
pin  P.  The  index  pin  H  is 
adjustable  and,  by  the  tension 
of  a  spring,  is  held  in  any 
one  of  the  holes  in  the  index 
plate.  The  sector  arms  /,  / 
can  be  set  to  include  any 


MECHANISM    OF    DIVIDING    HEAD 


A,  spindle;  B,  index  plate;   C,  worm 
wheel ;  D,  single-thread  worm ;  E,  shaft; 
F,  index  crank;    H,  index  pin;    J,  /, 
sector  arms ;  P,  stop  pin 


desired  number  of  holes,  thus 
rendering  it  unnecessary  to 
count  the  holes  every  time  that  the  index  crank  is  turned. 

In  direct  indexing  the  worm  is  thrown  out  of  mesh,  and  the  spindle 
is  locked  in  the  desired  position  by  a  special  stop  pin. 


64 


INDEXING  AND  SPIRAL  CUTTING 


Simple  Indexing.  For  simple  indexing  we  make  use  of  the 
dividing  head  already  shown  on  page  63  and  here  represented 
again  in  detail. 

The  worm  wheel  (7,  which 
has  40  teeth,  is  driven  by  the 
single-thread  worm  Z>,  on  shaft 
E.  Therefore  every  revolu- 
tion of  the  worm  D,  which 
corresponds  to  a  single  revo- 
lution of  the  index  crank  F, 
moves  the  worm  wheel  C  ex- 
actly one  tooth  and  moves  the 


MECHANISM    OF    DIVIDING    HEAD 

A,  spindle ;  B,  index  plate ;   (7,  worm 


Spindle   A1    which    is    attached     wheel;  D,  single-thread  worm;  E,  shaft; 

fn  if      1     nf  a   rpvnlntinn  F'  index  Crank5    H>  index  Pin5    7»  *» 

tO  It,   ¥7  01  a  revolution.  sector  arms;  P,  stop  pin 

Therefore  40  revolutions  of 

the  index  crank  F  cause  the  spindle  A,  which  carries  the 
work,  to  make  one  revolution.  Hence  if  40  divisions  are 
required  on  the  work,  one  revolution  of  F  must  be  made 
between  each  pair  of  cuts;  if  20  divisions  are  required,  two 
revolutions  of  F  are  necessary ;  if  10  divisions  are  required, 
four  revolutions  of  F  are  necessary ;  and  so  on. 

Formula  for  Simple  Indexing.  Letting  D  be  the  number 
of  divisions  required  and  C  the  number  of  revolutions  of  the 
index  crank,  we  have  the  following  simple  formula: 


For  example,  using  the  above  instrument,  find  the  indexing  required 
to  obtain  80  divisions. 

C  =  —  =  —  =  - 

D      80      2* 


We  have 


Therefore  any  circle  of  holes  in  the  index  plate  may  be  used  pro- 
vided the  number  of  holes  is  divisible  by  2.  In  this  case  we  might  take 
a  16-hole  circle  and  move  the  index  crank  8  holes  for  each  division. 


SIMPLE  INDEXING  65 

Index  Plates.  Different  makes  of  dividing  heads  have  dif- 
ferent sets  of  index  plates.  In  our  exercises  we  shall  use 
plates  which  have  circles  with  the  number  of  holes  as  follows  : 

BROWN  AND  SHARPE 

Plate  1 :  15,  16,  17,  18,  19,  20 
Plate  2 :  21,  23,  27,  29,  31,  33 
Plate  3:  37,  39,  41,  43,  47,  49 

CINCINNATI  MILLING  MACHINE  CO. 
One  plate  drilled  on  both  sides,  as  follows: 
First  side:       24,  25,  28,  30,  34,  37,  38,  39,  41,  42,  43 
Second  side:  46,  47,  49,  51,  53,  54,  57,  58,  59,  62,  66 

Exercises.    Simple  Indexing 

1.  If  a  spur  gear  is  to  have  36  teeth,  find  the  indexing 
required  on  a  Cincinnati  dividing  head. 

Since  C  =  f  §  =  1^,  we  must  have  a  circle  with  the  number  of  holes 
divisible  by  9,  in  this  case  54.  The  indexing  required  is  therefore  one 
revolution  of  the  index  crank  plus  6  holes  on  the  54-hole  circle. 

2.  Find  the  indexing  required  on  a  B.  &  S.  (Brown  and 
Sharpe)  dividing  head  for  7  divisions. 

Find  the  indexing  on  a  B.  $  S.  dividing  head  for  each  of 
the  following  divisions  : 

3.  3.            5.  45.  7.  98.             9.  145.          11.  205. 

4.  26.           6.  68.  8.  116.   '        10.  164.           12.  312. 

Find  the  indexing  on  a  Cincinnati  dividing  head  for  each  of 
the  following  divisions : 

13.  5.  15.  17.  17.  44.  19.  70.  21.  92. 

14.  9.  16.   23.  18.   55.  20.  85.  22.  110. 


66  INDEXING  AND  SPIRAL  CUTTING 

Angular  Indexing.  If  the  dividing  head  requires  40  turns 
of  the  index  crank  for  one  revolution  of  the  spindle,  that  is, 
to  turn  the  work  through  360°,  one  turn  of  the  crank  will 
produce  a  turning  of  -^  of  360°,  or  9°.  The  process  of 
indexing  work  by  angles  is  known  as  angular  indexing. 

Therefore,  if  one  revolution  of  the  index  crank  gives  an 
angle  of  9°,  two  holes  on  the  18-hole  circle  or  three  holes  on 
the  27-hole  circle  of  a  B.  &  S.  dividing  head  will  correspond 
to  1°.  Halves  and  thirds  of  a  degree  are  indexed  by  taking 
one  hole  on  the  18-hole  circle  or  one  hole  on  the  27-hole 
circle  respectively.  On  a  Cincinnati  dividing  head  the  54-hole 
circle  is  used,  six  holes  corresponding  to  1°,  and  so  on. 

For  example,  using  a  B.  &  S.  dividing  head,  find  the  indexing  for  an 
angle  of  25°. 

Since  one  turn  of  the  index  crank  gives  an  angle  of  9°,  the  number 
of  turns  of  the  crank  will  be  -^5-,  or  2£.  The  indexing  required  is,  there- 
fore, two  revolutions  of  the  crank  plus  14  holes  on  the  18-hole  circle. 

Exercises.    Angular  Indexing 

1.  Find  the  indexing  on  a  B.  &  S.  head  for  an  angle  of  11°. 

2.  Find  the  indexing  on  a  Cincinnati  dividing  head  for 


an  angle  of 

Using  a  B.  $  S.  dividing  head,  find  the  indexing  for  each 
of  the  following  angles  : 

3.  1°.  5.  4J°.  7.  13°.  9.  33°.          11.  44f°. 

4.  2J°.         6.   8f°.  8.  17i°.        10.  40i°.        12.  451°. 

Using  a  Cincinnati  dividing  head,  find  the  indexing  for  each 
of  the  following  angles  : 

13.  3°.         15.  11°.          17.   23°.          19.  56^°.        21.   79|°. 

14.  51°.       16.  19|°.        18.   31  J°.       20.   75°.          22.   82  '°. 


ANGULAR  AND  COMPOUND  INDEXING  67 

Compound  Indexing.  To  obtain  divisions  which  are  beyond 
the  range  of  simple  indexing  a  device  known  as  compound 
indexing  is  sometimes  employed. 

In  this  method  the  index  crank  is  first  turned  a  definite 
amount  in  the  regular  way,  and  then  the  stop  pin  holding 
the  index  plate  is  disengaged  and  the  index  plate  is  turned 
either  in  the  same  direction  or  in  the  opposite  direction.  The 
indexing  is  thus  compounded  of  two  separate  movements 
which  are,  in  reality,  two  simple-indexing  operations. 

Compound  indexing  should  be  used  only  when  there  is  no  other  con- 
venient way  of  obtaining  the  division.  This  is  because  of  the  chances  of 
error  due  to  the  fact  that  the  holes  must  be  counted  when  moving  the 
index  plate,  as  will  be  seen  in  the  example  below.  The  method  has  been 
largely  superseded  by  the  differential  method  explained  on  page  70. 

If,  for  example,  the  index  crank  of  a  B.  &  S.  dividing  head 
is  turned  40  holes  on  the  18-hole  circle,  that  is,  two  revo- 
lutions plus  four  holes,  and  then,  the  stop  pin  being  disen- 
gaged, the  index  plate  is  turned  40  holes  on  the  19-hole  circle, 
that  is,  two  revolutions  plus  two  holes,  in  the  same  direction, 
the  two  movements  will  cause  the  spindle  holding  the  work 
to  make  ^  +  Jg,  or  -f^  of  a  revolution. 

If,  however,  the  index  plate  is  turned  in  the  opposite 
direction  from  the  index  crank,  the  spindle  will  make  y1^  —  Jg, 
or  g^,  of  a  revolution.  Since  the  two  revolutions  of  the 
index  crank  and  the  index  plate  are  in  opposite  directions, 
they  offset  each  other,  and  the  same  result  is  obtained  by 
moving  the  index  crank  four  holes  on  the  18-hole  circle  in 
one  direction  and  the  index  plate  two  holes  on  the  19-hole 
circle  in  the  opposite  direction. 

Thus,  by  combining  the  two  movements  we  can  obtain 
342  divisions  with  the  index  plates  listed  on  page  65.  With 
simple  indexing  we  should  have  required  a  plate  with  a 
circle  of  171  holes. 


68  INDEXING  AND  SPIRAL  CUTTING 

Illustration  of  Compound  Indexing.  For  example,  to  find 
the  indexing  required  on  a  B.  &  S.  dividing  head  to  obtain 
147  divisions,  we  proceed  as  follows: 

1.  Find  two  factors  of  147. 
That  is,  147  =  3  x  49. 

2.  Try  to  find  two  circles  on  the  same  plate,  one  with  49 
holes  and  the  other  with  perhaps  ten  holes  less.    In  this  case 
let  us  try  the  49-hole  and  39-hole  circles. 

3.  Factor  the  difference  between  the  numbers  of  holes  in 
the  two  circles. 

That  is,  49  -  39  -  10  =  2  x  5. 

4.  Take  the  indicated  product  of  these  two  sets  of  factors 
as  the  numerator  of  a  fraction. 

That  is,  the  numerator  is  (3  x  49)  x  (2  x  5). 

5.  Take  for  the  denominator  of  this  fraction  (1)  the  fac- 
tors of  40,  that  is,  of  the  number  of  turns  of  the  index  crank 
to  one  turn  of  the  dividing-head  spindle ;  (2)  the  factors  of 
the  larger  circle ;  (3)  the  factors  of  the  smaller  circle. 

That  is,  (1)     40  =  2  x  2  x  2  x  5, 

(2)  49  =  7  x  7, 

(3)  39  =  3  x  13, 

and  hence  the  denominator  is  (2  X  2  X  2  X  5)  X  (7  X  7)  x  (3  x  13). 

6.  Now  cancel,  and  if   all  the  factors  in  the  numerator 
cancel  out,  the  correct  circles  have  been  chosen. 

7 
_      ,  .  ,  (48  x  3")  X  (2  X  0) 

In  this  case  we  have     ^- E2 — ?£• M - — , 

(2  x  2  x  £  x  0)  x  (/  x  7)  x  (JJ  x  13) 

and  therefore  the  correct  circles  have  been  chosen. 

If  all  the  factors  in  the  numerator  do  not  cancel  out,  other  numbers 
representing  the  numbers  of  holes  in  the  circles  of  the  index  plates  must 
be  chosen  until  two  circles  are  found  that  will  permit  the  cancellation 
of  all  these  factors. 


COMPOUND  INDEXING  69 

7.  Take  the  product  of  the  uncanceled  factors  in  the 
denominator  as  the  numerator  of  each  of  two  fractions,  called 
the  indexing  fractions,  in  which  the  numerator  is  the  number 
of  holes  to  be  turned,  and  the  denominator  is  the  number  of 
holes  in  the  circle  on  which  the  number  turned  is  counted. 

In  this  case  the  numerator  of  the  indexing  fractions  is  2  x  2  x  13, 
or  52,  and  the  fractions  which  give  the  desired  divisions  are  +  ^f 
and  —  |§,  the  signs  +  and  —  being  used  to  show  that  the  two  move- 
ments are  to  be  made  in  opposite  directions. 

That  is,  by  turning  52  holes  on  the  39-hole  circle  in  one  direction 
and  52  holes  on  the  49-hole  circle  in  the  other  direction,  we  shall  have 
the  proper  movement  of  the  spindle. 

This  is  seen  from  the  fact  that 


If  any  whole  number  is  subtracted  from  each  of  two  fractions,  the 
difference  of  the  fractions  is  not  affected.  In  this  case,  therefore,  we 
may  subtract  1  from  each  fraction,  and  we  have 

«-l=49,  and  »-!=&• 

Hence  we  give  the  indexing  fractions  required  to  obtain  147  divisions 
by  compound  indexing  as  +  ^f  and  —  ^. 

Exercises.    Compound  Indexing 

Using  compound  indexing  on  a  B.  $  S.  dividing  head,  find 
the  indexing  fractions  for  each  of  the  following  divisions  : 

1.  69.  2.  51.  3.  99.  4.  217.  5.  294. 

6.  By  the  compound  method,  find  the  indexing  fractions 
for  cutting  a  spur  gear  'with  77  teeth  on  a  milling  machine 
which  is  equipped  with  a  B.  &  S.  dividing  head. 

7.  Find  the  fractions  in  Ex.  6  for  a  gear  with  87  teeth. 

8.  Find  the  indexing  fractions   for  cutting  129  teeth  on 
the  rim  of  a  ratchet  wheel,  using  compound  indexing  on  a 
B.  &  S.  dividing  head. 


70 


INDEXING  AND  SPIRAL  CUTTING 


Differential  Indexing.  The  method  of  indexing  known  as 
differential  indexing  is  the  same  in  general  principle  as  com- 
pound indexing.  It  differs,  however,  from  the  latter  in  that 
the  index  plate  is  revolved  by  a  suitable  gearing  interposed 
between  it  and  the  dividing-head  spindle,  the  stop  pin  holding 
the  index  plate  being  disengaged  altogether.  The  movement 


DIVIDING   HEADS    GEARED   FOR    DIFFERENTIAL    INDEXING 

The  index  plate  of  the  dividing  head  shown  at  the  left  is  connected  to  the  spindle 
by  simple  gearing ;  that  of  the  dividing  head  shown  at  the  right,  by  compound 
gearing.  The  letters  show  the  following  parts :  P,  stop  pin  ;  R,  gear  on  spindle 
(driving) ;  S,  gear  on  bevel-gear  shaft,  or  worm,  (driven) ;  T,  intermediate  gear ; 
U,  first  gear  on  stud  (driving) ;  V,  second  gear  on  stud  (driven) 

of  the  index  plate  takes  place  when  the  crank  is  turned, 
the  index  plate  moving  either  in  the  same  direction  as  the 
crank  or  in  the  opposite  direction. 

Therefore,  in  using  differential  indexing  to  obtain  a  given 
number  of  divisions  we  have  to  find  two  factors,  (1)  the 
indexing  fraction  and  (2)  the  change  gears  which  will  turn 
the  index  plate  the  required  amount  in  the  proper  direction 
during  the  movements  of  the  index  crank. 

Change  Gears.    The  change  gears  furnished  with  a  B.  &  S. 

dividing  head  for  differential  indexing   and  spiral  milling 

have  the  following  numbers  of  teeth: 

24     24     28     32     40     44    48     56     64     72    86     100 
In  solving  the  exercises  on  differential  indexing  use  only  these  gears. 


DIFFERENTIAL  INDEXING 


71 


Illustration  of  Differential  Indexing.  For  example,  let  us 
consider  the  differential  indexing  on  a  B.  &  S.  dividing  head 
for  59  divisions. 

In   simple    indexing  p^=»  Cj 

for  60  divisions  the 
movement  of  the  index 
crank  is  |$,  or  J,  of  a 
turn  for  each  cut. 

If  the  crank  is  given 
|  of  a  turn  59  times, 
it  makes  391  turns, 
or  |  of  a  turn  less  than 
the  40  turns  required 
for  one  revolution  of 
the  work.  Hence  the 
index  plate  must  move 
in  the  same  direction 
as  the  crank  ^  of  a 

B 

revolution     while     the 
work  revolves  once. 
We  therefore  have 


59  x  §  =  39^, 

and     40-39J  =  f. 

Hence  the  ratio  of  the 

gears  is  §  to  1,  or  2  : 3, 
16  x  2  ^  32 
16  x  3  ~48' 


and 


MECHANISM    OF    DIVIDING    HEAD    GEARED 
FOR    DIFFERENTIAL    INDEXING 

A,  spindle;  B,  index  plate;  C,  worm  wheel; 
D,  worm  ;  E,  shaft ;  H,  index  pin ;  P,  stop  pin  ; 
ft,  gear  on  spindle  (driving) ;  S,  gear  on  bevel- 
gear  shaft,  or  worm,  (driven) ;  !T,  intermediate 
gear ;  U,  first  gear  on  stud  (driving) ;  V,  second 
gear  on  stud  (driven) 


A  32-T  gear  (driving)  is  placed  on  the  special  differential-indexing 
center  in  the  spindle  of  the  dividing  head,  and  a  48-T  gear  (driven) 
is  placed  on  the  bevel-gear  shaft  which  turns  the  index  plate.  An 
intermediate  gear  is  used  to  make  the  index  plate  move  in  the  same 
direction  as  the  index  crank,  which  is  given  §  of  a  turn  for  each  cut ; 
that  is,  the  indexing  fraction  may  be  given  as  -^f . 


72  INDEXING  AND  SPIRAL  CUTTING 

Exercises.    Review 

1.  Find  the  differential  indexing  on  a  B.  &  S.  dividing 
head  for  175  divisions. 

For  180  divisions,  T4g°o-  =  §• 

Then  175  x  f  =  38f ,     and     40-38f  =  l^. 

Hence  the  required  gearing  must  be  in  the  ratio  of  1£  to  1,  or  10  :  9. 
Since  no  two  gears  will  give  this  ratio,  we  have 

JQ.  40   S    y    5 

D     —  US'  —  54" 

Gears  in  the  ratio  of  8  (driving)  to  9  (driven)  are  used  on  the  spindle 
and  bevel-gear  shaft,  and  gears  in  the  ratio  of  5  (driving)  to  4  (driven) 
are  used  on  the  stud.  We  therefore  have 

8_x_8  =  64  8  x  5  _  40 

8  x  9  ~  72 '  8  x  4  ~  32 ' 

The  arrangement  of  the  gears  is  as  follows :  spindle,  64  T ;  bevel- 
gear  shaft,  72  T;  first  gear  on  stud,  40  T;  second  gear  on  stud,  32  T. 
With  compound  gearing  an  intermediate  gear  is  not  needed  to  make 
the  index  plate  turn  in  the  same  direction  as  the  index  crank.  The  in- 
dexing fraction  may  be  given  as  T4?,  which  is  equivalent  to  f  of  a  turn. 

Find  the  differential  indexing  on  a  B.  $  S.  dividing  head 
for  each  of  the  following  divisions  : 

2.  53.           4.  107.           6.  161.  8.  199.           10.  271. 

3.  67.           5.  112.           7.  173.  9.  214.           11.  321. 

Using  a  Cincinnati  dividing  head,  find  the  simple  indexing 
movement  for  each  of  the  following  divisions : 

12.  116.         13.  145.         14.  205.         15.  245.         16.  392. 

Using  a  B.  $  S.  dividing  head,  find  the  indexing  movement 
for  each  of  the  following  angles : 

17.  12°.        18.   211°.        19.  291°.       20.   53|°.       21.  66£°. 

22.  Using  compound  indexing  on  a  B.  &.  S.  head,  find  the 
indexing  fractions  for  milling  273  slots  on  a  feed  disk. 


SPIRALS  73 

Spirals.  A  winding  cut  made  by  a  tool  moving  around 
a  cone  and  at  the  same  time  advancing  along  its  slant  line 
at  a  uniform  rate  is  called  a  spiral  cut. 

Since  the  method  of  feeding  the  work  is  the  same  in  the  case  of  a 
straight  reamer  as  in  that  of  a  taper  reamer,  it  is  customary  to  give  the 
name  "  spiral  cut "  to  the  helical  flutes  of  straight  drills  and  reamers  and 
also  to  the  spiral  teeth  in  mills,  gears,  and  cutters. 

The  dividing  head  is  used  not  only  for  indexing  but  also 
for  the  milling  of  spirals.  When  a  spiral  is  being  milled  the 
work  is  turned  slowly  by  the  dividing  head,  while  the  table 
of  the  milling  machine  feeds  lengthwise. 

The  lead  of  a  spiral  is  figured  in  much  the  same  way  as 
the  lead  of  a  screw  thread ;  that  is,  it  is  the  distance  that  the 
spiral  advances  in  making  one  turn  around  the  work.  The 
spiral,  however,  is  spoken  of  as  having  so  many  inches  to 
one  turn  instead  of  as  having  so  many  threads  to  one  inch, 
as  is  the  case  with  a  screw  thread. 

In  the  process  of  milling  a  spiral,  change  gears  are  used  to 
connect  the  dividing  head  with  the  feed  screw  on  the  table 
of  the  milling  machine,  so  that  the  spindle  of  the  dividing 
head  rotates  a  definite  amount  in  proportion  to  the  distance 
that  the  table  travels.  The  action  of  this  gearing  is  similar  to 
that  used  in  cutting  a  thread  in  a  lathe. 

Spirals  are  always  cut  on  a  universal  milling  machine, 
the  table  of  which  can  be  set  at  any  desired  angle  to  the 
spindle  holding  the  milling  cutter.  The  table  is  set  at  such 
an  angle  as  will  bring  the  milling  cutter  in  line  with  the  cut 
to  be  made  on  the  work,  as  is  explained  on  page  78.  The 
angle  at  which  the  table  is  set  has  no  effect  on  the  lead  of 
the  spiral,  which  is  determined  by  the  change  gears  used. 
If  the  table  were  left  in  its  normal  position  at  right  angles 
to  the  cutter  spindle,  a  spiral  of  the  same  lead  would  be 
obtained,  but  the  shape  of  the  cut  would  be  changed. 


74 


INDEXING  AND  SPIRAL  CUTTING 


Spiral  Milling.  The  figure  below  shows  both  the  side 
view  and  the  end  view  of  a  dividing  head  mounted  on  the 
table  of  a  milling  machine  and  arranged  for  spiral  milling. 

The  rotary  movement  of  the  dividing-head  spindle  A,  which 
drives  the  work,  is  caused  by  the  turning  of  the  feed 
screw  B,  which  also  causes  the  table  to  move  longitudinally. 


40 


DIVIDING   HEAD   CONNECTED   BY   COMPOUND    GEARING  WITH   FEED 
SCREW  OF   MILLING   MACHINE 

A,  spindle ;  B,  feed  screw ;  (7,  first  gear  on  stud  (driving) ;  D,  gear  on  bevel-gear 

shaft,  or  worm,  (driven) ;  E,  second  gear  on  stud  (driven) ;  F,  gear  on  feed  screw 

(driving) ;  P,  stop  pin;  R,  bevel-gear  shaft;  T,  teeth,  as  in  32 T 

The  motion  of  the  feed  screw  B  is  transmitted  through 
the  compound  train  of  change  gears  F,  E,  (7,  D  to  the  bevel- 
gear  shaft  R,  which  has  a  set  of  bevel  gears  inside  the  divid- 
ing head  connecting  with  the  shaft  on  which  the  worm 
which  drives  the  spindle  is  cut,  as  will  be  seen  in  the  figures 
on  pages  63  and  71.  The  gear  on  the  feed  screw  is  custom- 
arily spoken  of  as  the  "  gear  on  screw,"  while  that  on  shaft 
R  is  called  the  "  gear  on  worm."  When  the  dividing  head 
is  used  for  spiral  milling,  the  stop  pin  P,  which  holds  the 
index  plate,  is  disengaged  altogether. 


SPIKAL  MILLING  75 

Lead  of  a  Milling  Machine.  If  the  change  gears  are  in  the 
ratio  of  1  to  1  (that  is,  if  gear  D  revolves  once  for  each  revo- 
lution of  F),  the  lead  of  the  milling  machine  is  the  distance  that 
the  table  travels  while  the  spindle  revolves  once. 

This  distance  is  a  constant  used  in  figuring  change  gears  and  varies 
with  different  makes  of  machines. 

If  R  is  the  number  of  revolutions  of  the  feed  screw  for 
one  revolution  of  the  spindle  with  1-to-l  gearing  and  F  is 
the  lead  of  the  feed  screw,  Jf,  the  lead  of  the  milling  machine, 
is  found  as  follows : 

M  =  RF 

For  example,  if  with  1-to-l  gearing  R  =  40  on  a  milling  machine 
which  has  F  =  £",  we  have 

3/^/^  =  40  x  4"  =  10". 

Change  Gears.  Letting  L  be  the  lead  of  the  spiral  to  be 
cut  and  M  the  lead  of  the  milling  machine,  the  ratio  of 
driven  to  driving  gears  is  found  from  the  following  formula  : 

Gears  driven       L 
Gears  driving  ~~  M 

Letting  Tw  represent  the  gear  on  worm  (driven),  Ts  the 
gear  on  screw  (driving),  Sl  the  first  gear  on  stud  (driving), 
and  S2  the  second  gear  on  stud  (driven),  the  above  ratio  is 
split  into  two  factors  as  follows: 

Gears  driven  _  T 
Gears  driving  ~~  Ts 

The  terms  of  each  of  the  ratios  Tw :  T8  and  S2 :  Sl  are  multiplied  by 
a  number  which  will  give  numbers  in  the  list  of  gears  on  page  70. 

To  find  the  lead  of  a  spiral  which  will  be  cut  with  a  given 
combination  of  change  gears  we  have  the  following  formula : 
_  M  x  product  of  driven  gears 
product  of  driving  gears 


76  INDEXING  AND  SPIKAL  CUTTING 

Illustrative  Problems.  1.  Find  the  change  gears  required 
on  a  milling  machine,  which  has  a  lead  of  10",  for  cutting  a 
spiral  with  a  lead  of  36"- 

Using  the  formulas  given  on  page  75,  we  have 

Gears  driven  _  L  _  36  _  Tw  ^  £2  _  9  ^  4 
Gears  driving  ~  M  ~  10  ~~  T8  *  Sl  ~  5  X  2  ' 

Tw      9      8  x  9      72  ,     52      4      16  x  4      64 

whence     _^-  =  _  =  _,     and     -2  =  -  =  __  =  _. 

We  therefore  have  the  following  arrangement  of  gears :  gear  on 
worm  (driven),  72  T  ;  gear  on  screw  (driving),  40  T  ;  first  gear  on  stud 
(driving),  32  T  ;  and  second  gear  on  stud  (driven),  64  T. 

2.  Find  the  change  gears  required  on  a  milling  machine 
with  a  lead  of  10"  for  cutting  a  spiral  with  a  lead  of  12.80". 

Gears  driven  =  L  =  12.80  _  128  =  Tw      S^  _  _6£      2 
Gears  driving      M  ~     10     ~  100  ~  T8  *  ^  ~~  100  X  1 ' 

Tw       64  5«      2      28  x  2      56 

Then  — -  = >     and     — -  =  —  = =  —  • 

T8      100  Sl      1      28  x  1      28 

We  therefore  have  the  following  arrangement  of  gears:  gear  on 
worm  (driven),  64  T  ;  gear  on  screw  (driving),  100  T  ;  first  gear  on  stud 
(driving),  28  T  ;  and  second  gear  on  stud  (driven),  56  T. 

3.  What  is  the  lead  of  the  spiral  that  will  be  cut  on  a 
milling  machine  which  has  a  lead  of  10"  when  the  following 
arrangement  of  change  gears  is  used :  gear  on  worm,  86  T ; 
first  gear  on  stud,  40  T;  second  gear  on  stud,  48  T  ;  and  gear 
on  screw,  44  T  ? 

Using  the  formula  given  at  the  foot  of  page  75,  we  have 

3 

» 

T  _M  X  product  of  driven  gears  _  ffl  x  86  x  $fi  _  258  _  _ 

product  of  driving  gears  £0  x  ££  11 

t       11 

Hence  the  lead  of  the  spiral  that  will  be  cut  is  23.45". 


CUTTING  SPIRALS  77 

Exercises.    Cutting  Spirals 

1.  If  a  milling  machine  has  a  lead  of  10",  what  change 
gears  are  needed  to  cut  a  spiral  with  a  lead  of  16"  ? 

2.  If  a  milling  machine  has  a  lead  of  10",  what  change 
gears  are  needed  to  cut  a  spiral  with  a  lead  of  22.50"  ? 

3.  For  milling   a  spiral  the  change  gears  on   a  milling 
machine  which  has  a  lead  of  10"  were  placed  as  follows:  a 
64-T  gear  as  the  gear  on  worm,  a  40-T  gear  as  the  gear  on 
screw,  a  32-T  gear  as  the  first  gear  on  stud,  "and  a  56-T  gear 
as  the  second  gear  on  stud.    What  is  the  lead  of  the  spiral 
which  was  cut  with  this  arrangement  of  gears  ? 

Find  the  change  gears  required  on  a  milling  machine  with  a 
lead  of  10"  for  milling  each  of  the  following  spiral  leads  : 

4.  0.80".  7.  4.125".  10.  18.75".          13.  31.50". 

5.  2.20".  8.  12.60".  11.  21.00".  14.  37.50". 

6.  3.50".  9.  17.20".  12.  25.80".          15.  52.50". 

16.  Find  the  change  gears  required  on  a  milling  machine 
with  a  lead  of  10"  in  cutting  a  spiral  of  one  turn  in  13J". 

17.  In  milling  a  spiral  the  following  arrangement  of  change 
gears  was  used :  gear  on  worm,  86  T ;  gear  on  screw,  56  T ; 
first  gear  on  stud,  28  T ;  second  gear  on  stud,  48  T.    If  the 
lead  of  the  milling  machine  was  20",  what  was  the  lead  of 
the  spiral  which  was  milled  ? 

18.  What  is  the  lead  of  the  spiral  which  will  be  cut  on  a 
milling  machine  with  a  lead  of  10"  when  there  is  a  64-T 
gear  on  the  bevel-gear  shaft,  or  worm,  a  32-T  gear  as  the  first 
gear  on  the  stud,  a  48-T  gear  as  the  second  gear  on  the  stud, 
and  a  56-T  gear  on  the  feed  screw  ? 

19.  What  would  be  the  lead  of  the  spiral  in  Ex.  18  if  the 
gears  on  the  stud  were  reversed? 


78 


INDEXING  AND  SPIRAL  CUTTING 


Setting  the  Table.  When  cutting  a  spiral  the  table  of  the 
milling  machine  is  set  at  the  angle  which  the  spiral  cut  is  to 
make  with  the  axis  of  the  work,  as  explained  on  page  73. 


MILLING-MACHINE    TABLE   SET   TO   ANGLE    FOR   SPIRAL   CUTTING 

The  position  of  the  table  shown  by  A  is  for  cutting  a  right-hand  spiral,  and  that 
shown  by  B  is  for  cutting  a  left-hand  spiral 

As  shown  in  the  figures  below,  there  is  a  fixed  relation 
connecting  the  spiral  angle  #,  the  lead  Z,  and  the  circumfer- 
ence of  the  work  C. 

i *1 

If  we  let  the  base  of  a 

right  triangle  be  equal 

to   the   lead  £,    as  shown  in   the 

second  figure,  and  let  the  altitude 

be  equal  to  the  circumference  C,  the 

angle  a  formed  by  the  hypotenuse 

and  the  base  is  the  required  angle  at  which  to  set  the  table 

of  the  milling  machine,  and  is  found  by  the  use  of  tangents. 


SETTING  THE  TABLE  79 

Illustrative  Problem.  Find  the  angle  at  which  to  set  the 
table  of  a  milling  machine  when  cutting  a  spiral  with  a  lead 
of  28.00"  on  a  3i-inch  blank. 

Referring  to  the  lower  figures  on  page  78,  in  which  C  represents  the 
circumference  of  the  work  and  L  the  lead  of  the  spiral,  we  see  that 

tan  a  =  —  • 
L 

C      3.1416  x  31 
Then  tan  a  —  -  = — ±  =  0.3647, 

and  hence  a  =  20°  2'. 

Since  the  scale  by  which  angle  a  is  set  is  usually  graduated  only  to 
fourths  of  a  degree,  we  give  the  above  result  as  20°. 


Exercises.    Setting  the  Table 

Find  the  angle  at  which  to  set  the  table  of  a  milling  machine 
to  cut  each  of  the  following : 

1.  A  spiral  with  a  lead  of  13.12"  on  a  spiral-gear  blank 
which  is  2|"  in  diameter. 

In  all  problems  dealing  with  setting  the  table  of  a  milling  machine 
give  the  results  to  the  nearest  J°. 

2.  A  spiral  of  8.95-inch  lead  on  a  1-inch  twist  drill. 

3.  A  spiral  with  a  lead  of  15.75"  on  a  milling  cutter  which 
is  to  be  4"  in  diameter. 

o 

4.  A  spiral  with  a  lead  of  0.67"  on  a  |-inch  spindle. 

5.  A  spiral  with  a  lead  of  67.45"  on  a  spiral-gear  blank 
which  has  a  diameter  of  5|". 

6.  If  the  angle  of  the  spiral  on  a  twist  drill  \\n  in  diameter 
is  191°,  what  is  the  lead? 

7.  If  the  table  of  a  milling  machine  is  set  at  an  angle  of 
10^°  for  cutting  a  spiral  with  a  lead  of  68.57",  what  is  the 
diameter  of  the  work  ? 


80 


INDEXING  AND  SPIRAL  CUTTING 


0.87 


Cams.  A  rotating  part  of  a  machine  that  gives  a  recipro- 
cating or  oscillating  motion  to  another  part  is  called  a  cam. 

The  piece  to  which  the  motion  is  thus  given  is  called  the  follower,  and 
the  contact  between  the  two  is  called  the  line  of  contact. 

A  cam  may  also  have  a  reciprocating  or  oscillating  motion. 

The  distance  that  the  follower  is  moved  by  the  action  of 
the  cam  is  called  the  rise,  or  lead,  of  the  cam.  Thus,  the 
cam  shown  in  this  figure 
has  a  constant  rise  in  0.87 
of  its  circumference,  and  in 
one  revolution  imparts  to 
the  follower  a  movement 
of  0.150". 

Cams  are  made  in  many 
different  shapes,  and  often 
have  more  than  one  lobe, 
but  we  shall  consider  only 
what  are  known  as  plate 
cams,  or  peripheral  cams. 

Plate  cams  having  a 
constant  rise,  such  as  are 
used  on  automatic  screw 
machines,  can  be  cut  on  a 
universal  milling  machine. 

Since  these  cams  usually  have  shorter  leads  than  can  be 
obtained  on  a  milling  machine  by  any  practical  combination 
of  change  gears,  it  is  necessary  when  cutting  them  to  use  the 
vertical-spindle  attachment  of  the  milling  machine  to  hold 
the  milling  cutter. 

Such  cams  are  required  in  great  variety,  each  differing  from 
the  others  by  only  a  few  thousandths  of  an  inch.  Since 
there  are  practically  no  duplications,  it  is  impracticable  to 
cut  them  by  means  of  master  cams. 


0.6 


0.4 


A    CAM    OF    ONE    LOBE 

The  lobe  extends  through  0.87  of  the  cir- 
cumference,  or  approximately  313°,   and 
the  cam  has  a  lead  of  0.150" 


CAMS 


81 


Milling  Cams.    When   the   dividing  head  is  set  with  its 
spindle  at  right  angles  to  the  table  of  the  milling  machine, 
the  lead,  which  will  be  cut  on  a  cam  if  milled  for  a  complete 
revolution,  is  the  same  as  the  lead  of  the  spiral  cut  with  the 
same  change  gears.  If,  how- 
ever, the  dividing  head  and 
the  vertical-spindle  attach- 
ment are  both  set  at   an 
angle,   any   required    lead 
can  be  obtained,  provided 
it  is  less  than  the  spiral  lead 
for  which  the  milling  ma- 
chine   is    geared.      There- 
fore  change   gears,  which 
will    give    a    longer    lead 
than  that  to  be  cut  on  the 
cam,  are  first  selected  from 
a  table  of  spiral  leads. 

The  dividing  head  is  then 
elevated  from  the  horizon- 
tal a  certain  angle,  which 
is  found  by  « the  formula 
given  on  page  82,  and  the 
vertical-spindle  attachment, 
which  drives  the  milling 

cutter,  is  swiveled  around  from  its  vertical  position  an  amount 
equal  to  the  complement  of  the  angle  at  which  the  dividing 
head  is  set.  This  places  the  milling  cutter  in  line  with  the 
spindle  of  the  dividing  head  so  that  the  edges  of  the  cam 
will  be  cut  parallel  with  its  axis. 

Since  the  graduations  for  elevating  the  dividing  head  are  usually 
marked  only  in  fourths  of  a  degree,  in  all  problems  dealing  with  cams 
the  angles  are  to  be  figured  to  the  nearest  ^°. 


MILLING   A  SCREW-MACHINE   CAM 

£,  milling  cutter;  D,  vertical-spindle 
attachment ;   W,  cam  being  cut 


82 


INDEXING  AND  SPIRAL  CUTTING 


Finding  the  Angle.    If  A  is  the  angle  for  setting  the  divid- 
ing head,  c  the  lead  of  the  cam,  or  its  rise  if  continued  at  the 


DIVIDING    HEAD    SET    TO    ANGLE    FOR    MILLING    A    CAM 

A,  angle  of  dividing  head ;  B,  cutter ;  D,  vertical-spindle  attachment ;   W,  cam 

given  rate  for  a  complete  revolution,  and  s  the  lead  of  the 
spiral  for  which  the  milling  machine  is  geared,  we  have 

sin^  =  -. 
s 

If  the  rise  of  the  cam  is  given  only  for  n,  a  definite  num- 
ber of  hundredths  of  the  circumference,  the  formula  becomes 

c 

~  ns 

Illustrative  Problems.  1.  Find  the  angles  at  which  to  set 
the  dividing  head  and  the  vertical-spindle  attachment  in 
milling  a  cam  with  a  lead  of  0.602",  the  milling  machine 
being  geared  to  cut  a  spiral  with  a  lead  of  0.67". 

We  have 

and  hence  A  =  63°  58'. 

Calculating  to  the  nearest  £°,  the  dividing  head  is  elevated  64°,  and  the 
vertical-spindle  attachment  is  set  at  90°  —  64°,  or  26°,  from  the  vertical. 


CUTTING  PLATE  CAMS  83 

2.  In  Ex.  1  find  the  angles  if  the  cam  were  to  have  a  lead 
of  0.150"  in  300°,  or  in  0.83  of  the  circumference. 

We  have  sin  A  =  —  =  — 5lL52 —  =  0.2697, 

ns      0.83  x  0.67 

and  hence  A  =  15°  39'. 

The  dividing  head  is  elevated  15 f°  and  the  vertical-spindle  attach- 
ment is  swiveled  around  90°  —  15f °,  or  74^°,  from  the  vertical. 


Exercises.    Cutting  Plate  Cams 

1.  Find  the  angle  at  which  to  elevate  the  dividing  head 
when   cutting   a    cam    with  a    lead    of   1.249",  the    milling 
machine  being  geared  to  cut  a  spiral  with  a  lead  of  1.333". 

2.  A  milling  machine  is  geared  to  cut  a  spiral  with  a  lead 
of  1.860".    At  what  angles   should  the  dividing  head   and 
the   vertical-spindle    attachment  be   set   for   milling   a   cam 
with  a  lead  of  1.801"  ? 

3.  A  cam  having  a  rise  of  0.634"  in  0.75  of  its  circum- 
ference is  to  be  milled  on  a  milling  machine  geared  to  cut 
a  spiral  with  a  lead  of  0.930".    Find  the  angle  at  which  to 
elevate  the  dividing  head. 

4.  In  Ex.  3  find  the  angle  at  which  to  set  the   vertical- 
spindle  attachment  if  the  given  rise  were  to  be  obtained  in 
0.83  of  the  circumference  of  the  cam. 

A  milling  machine  being  geared  to  cut  a  spiral  with  a  lead 
of  0.916",  find  the  angle  at  which  to  elevate  the  dividing  head 
for  milling  a  cam  with  each  of  the  following  leads : 

5.  0.669".  8.  0.789".          11.  0.733".          14.  0.898". 

6.  0.691".  9.  0.712".          12.  0.859".          15.  0.771". 

7.  0.701".         10.  0.853".          13.  0.751".          16.  0.904". 


84  INDEXING  AND  SPIRAL  CUTTING 

Exercises.    Review 

1.  If  the  feed  screw  on  the  table  of  a  milling  machine 
has  a  lead  of  0.4"  and  it  requires  40  revolutions  of  the  feed 
screw  with  1-to-l  gearing  to  turn  the  dividing-head  spindle 
once,  what  is  the  lead  of  the  machine  ? 

2.  Find  the  change  gears  required  on  a  milling  machine 
with  a  lead  of  10"  for  milling  a  cutter  with  spiral  teeth  of 
9.60-inch  lead. 

3.  Find  the  angle  at  which  to  elevate  the  dividing  head 
when  milling  a  cam  with  a  rise  of  0.150"  in  0.75  of  the 
circumference,    the   machine   being   geared   to   cut   a   spiral 
with  a  lead  of  0.67". 

Find  the  angle  at  which  to  set  the  table  to  cut  a  spiral  with 
a  lead  of  13.760",  given  the  diameter  of  each  piece  as  follows : 

4.  1".  6.  \".  8.   f".  10.  1".  12.  |". 

5.  If".          7.  If".  9.   21".         11.  IJ".          13.   31£". 

14.  Find  the  lead  of  the  spiral  which  is  cut  on  a  milling 
machine  with  a  lead  of  10"  when  there  is  a  100-T  gear  on 
the  worm,  a  24-T  gear  as  the  first  and  a  64-T  gear  as  the 
second  gear  on  the  stud,  and  a  28-T  gear  on  the  screw. 

15.  Find  the  angles  at  which  to  set  the  dividing  head  and 
the  vertical-spindle  attachment  when  milling  a  cam  with  a 
rise   of  0.136"    in    0.85    of   the    circumference,   the   milling 
machine  being  geared  to  cut  a  spiral  with  a  lead  of  0.67". 

Find  the  change  gears  required  on  a  milling  machine  with  a 
lead  of  10"  for  milling  each  of  the  following  spiral  leads  : 

16.  0.90".          19.  3.75".         22.  12.375".          25.  27.50". 

17.  1.44".          20.  6.60".          23.  13.125".          26.  31.25". 

18.  2.45".          21.  8.25".          24.  16.875".          27.  42.00". 


CHAPTER  VI 


GEARS 

Nature  of  Gears.  If  two  wheels  are  mounted  on  parallel 
axes,  and  if  the  rims  of  the  wheels  are  placed  in  close  con- 
tact, when  one  is  turned  the  other  will  also  turn  if  there  is 
sufficient  friction.  The  surfaces  thus  placed  in  contact  are 
called  friction  surfaces. 

The  circumferences  of  these  wheels  are  called  pitch  circles, 
and  the  diameters  of  these  circles  are  called  pitch  diameters. 

To  prevent  slipping  the  friction  surfaces  are  usually 
grooved  so  as  to  form  teeth  which  mesh  together.  Wheels 
thus  provided  with  teeth  are  known  as  gear  wheels,  or  gears- 

A  toothed  wheel  in  which  the  teeth  are  cut  parallel  to  the 
axis  and  at  right  angles  to  the  sides  is  called  a  spur  gear. 

In  the  spur  gearing  here  shown 
the  smaller  gears  are  called  pinions 
and  the  straight  bar  is  called  a  rack. 

Besides  spur  gears  there  are 
gears  on  conical  surfaces  which 
transmit  power  to  shafts  at  an 
angle  to  each  other.  Such  gears  are 
known  as  bevel  gears.  Gears  having 
teeth  on  the  inside  of  the  rim  are 
called  internal  gears. 

Teeth.  It  is  evident  that  the 
teeth  may  have  various  shapes,  but  there  are  only  two  sys- 
tems in  general  use, —  the  involute,  or  single-curve  form  of 
tooth,  and  the  cycloidal,  or  double-curve  form.  The  cycloidal 
form  is  being  rapidly  replaced  by  the  involute  form. 

85 


SPUR   GEARING 


86  GEARS 

Terms  Used.  The  figure  on  page  87  illustrates  the  defini- 
tions of  the  following  terms,  which  we  shall  use  in  connection 
with  the  work  on  spur  gearing: 

Addendum.   The  height  (a)  of  a  tooth  above  the  pitch  circle. 

Center  distance.  The  distance  (^)c)  between  the  centers  of 
the  shafts  to  which  the  gears  are  fastened. 

Chordal  pitch.  The  distance  from  the  center  of  one  tooth 
to  the  center  of  the  next  tooth,  measured  on  a  straight  line; 
that  is,  on  the  chord  of  the  pitch  circle. 

Circular  pitch.  The  distance  (Pc)  between  the  centers  of 
two  adjacent  teeth,  measured  on  the  pitch  circle. 

It  includes  the  equivalent  of  one  space  and  one  tooth. 

Clearance.  The  distance  (c)  from  the  working-depth  circle 
to  the  whole-depth  circle  to  allow  freer  action  of  the  gears. 

Dedendum.  The  depth  (d)  of  a  tooth  between  the  pitch 
circle  and  the  working-depth  circle. 

Diametral  pitch.  The  number  of  teeth  per  unit  of  the  pitch 
diameter  of  the  gear. 

Thus,  if  there  are  48  teeth  on  a  gear  with  a  pitch  diameter  of  12", 
the  diametral  pitch  of  the  gear  is  Tf,  or  4. 

Unless  otherwise  stated,  the  "pitch"  of  a  gear  means  the  " diametral  pitch ." 

Face  of  a  tooth.  The  working  surface  (Fa)  outside  the  pitch 
circle  and  extending  the  width  of  the  tooth. 

Flank  of  a  tooth.  The  working  surface  (FF)  inside  the  pitch 
circle  and  extending  the  width  of  the  tooth. 

Outside  circle.   The  circumference  (  F)  of  the  gear  blank. 

Outside  diameter.    The  diameter  (I>0)  of  the  outside  circle. 

Pitch  circle.    The  circumference  of  the  friction  cylinder. 

Pitch  diameter.    The  diameter  (Dp)  of  the  pitch  circle. 

Pitch  line.    The  circumference  line  (X)  of  the  pitch  circle. 

Root.  The  total  depth  of  a  tooth  inside  the  pitch  circle, 
equivalent  to  the  dedendum  plus  the  clearance. 


TERMS  USED 


87 


Thickness  of  tooth.   The  thickness  (£)  of  the  tooth,  measured 
on  the  pitch  line  in  the  same  manner  as  the  circular  pitch. 


TERMS    USED    IN    SPUR    GEARING 

«,  addendum ;  c,  clearance ;  d,  dedendum ;  Dc,  center  distance ;  D0,  outside  diam- 
eter; Dp,  pitch  diameter;  Fa,  face;  Fl,  flank;  Pc,  circular  pitch ;  t,  thickness  of 
tooth  on  pitch  line ;   U,  whole-depth  circle ;    F,  working-depth  circle ;   W,  whole 
depth ;  X,  pitch  circle ;   F,  outside  circle 

Whole  depth.  The  distance  (JF)  from  the  outside  circle  to 
the  circle  (U)  formed  by  the  bottom  of  the  grooves. 

Working  depth.  The  distance  from  the  outside  circle  to  the 
circle  (F)  formed  by  the  top  of  the  teeth  of  the  mating  gear. 


88  GEARS 

Symbols.    We  shall  use  the  following  symbols  in  the  for- 
mulas and  computations  for  spur  gearing: 

a  =  addendum  Pc  —  circular  pitch 

c  =  clearance  Pd  =  diametral  pitch 

d  =  dedendum  r  =  root 

D0  =  outside  diameter  t  =  thickness  of  tooth 

Dv  =  pitch  diameter  W  =  whole  depth 

N=  number  of  teeth  W'  '  =  working  depth 

Spur-Gearing  Formulas.    The  following  formulas  used  in 
practical  work  may  be  referred  to  in  solving  the  exercises  : 


10  TT  D0 


Pd  P*  2 

7 
°~  7T 


(N+2)a  =Z>aP<f-2  W=2a+c 

N+2  TT  2 


=  3.1416 


N  N+2  TT 


SPUR-GEARING  FORMULAS  89 

Exercises.    Spur  Gears 

1.  A  gear  has  a  circular  pitch  of  0.7854".   Find  the  pitch. 

Remember  that  the  word  "  pitch  "  when  used  alone  always  refers  to 
the  diametral  pitch,  and  use  the  second  formula  for  Pd  on  page  88. 
The  difference  in  the  use  of  the  term  "  pitch  "  as  applied  to  gears  and 
to  screw  threads  should  be  noted.  Thus,  we  speak  of  a  screw  with 
4  threads  to  the  inch  as  a  screw  of  £-inch  pitch,  while  a  gear  which 
has  4  teeth  to  each  inch  of  the  pitch  diameter  is  a  4-pitch  gear. 

2.  What   is   the   circular   pitch  of   a  gear  which   has  a 
pitch  of  26? 

Carry  all  computations  for  dimensions  of  gears  to  four  decimal  places, 
but  in  the  final  result  discard  the  fourth  place,  giving  the  dimensions 
correct  to  the  nearest  0.001". 

3.  Find  the  outside  diameter  of  a  gear  having  70  teeth 
and  a  pitch  of  8. 

4.  Find  the  thickness  of  tooth  on  the  pitch  line  of  a  gear 
with  a  pitch  of  16. 

5.  A  gear  has  a  pitch  of  12  and  an  outside  diameter  of 
5".    Find  the  number  of  teeth. 

6.  If  Pc  =  2.5133",  what  is  the  value  of  Pd  ?  Write  a  state- 
ment of  the  meaning  of  the  problem,  not  using  the  symbols. 

The  value  of  Pd  should  be  given  as  1^. 

7.  If  Pd  =  9,  what  is  the  value  of  Pc  ?    Write  a  statement 
as  in  Ex.  6. 

8.  If  the  outside  diameter  of  a  gear  is  18"  and  the  pitch 
is  6,  what  is  the  number  of  teeth? 

9.  If  a  gear  has  24  teeth  and  a  pitch  of  18,  what  is  the 
outside  diameter? 

10.  What  should  be  the  diameter  of  the  gear  blank  for  a 
gear  which  is  to  have  46  teeth  and  the  circular  pitch  of 
which  is  to  be  0.1309"? 


90  GEAKS 

11.  Find  the  whole  depth  of  a  tooth  on  a  gear  having" 
a  pitch  of  10. 

It  will  be  seen  from  the  formulas  on  page  88  that  to  find  the  whole 

2 
depth  W  we  must  use  the  formula  W  =  •—  -  +  c>  in  which  we  find  c  from 

the  formula  c  -  —  and  t  from  the  formula  t  =  5-^-  •   Since  TT  =  3.1416, 

1  5708 
we  have  —  =  1.5708,  and  hence  t  =  —  -  .    If  we  substitute  this  value 

for  t  in  the  formula  c  =  —  -,  we  shall  then  have  c  =    '          ,  or  c  =  -^—  -- 

10  r&  -frf 

Using  this  value  for  c,  we  have 


Such  a  formula  when  once  developed  may  be  used  in  finding  the 
whole  depth  of  tooth  for  any  given  pitch. 

12.  How  many  teeth  should  there  be  on  a  gear  having  a 
pitch  of  12  and  an  outside  diameter  of  3|"? 

13.  If  _ZJ  =  1|,  what  is  the  value  of  Pc?  Write  a  statement 
of  the  meaning  of  the  problem,  not  using  the  symbols. 

14.  If  the  outside  diameter  of  a  gear  is  4"  and  the  number 
of  teeth  is  46,  what  is  the  pitch  ? 

15.  Find  the  number  of  teeth  for  a  gear  having  a  pitch  of 
14  and  an  outside  diameter  of  7f  ". 

16.  Find  the  thickness  of  a  tooth  on  an  8-pitch  gear. 

17.  Find  the  whole  depth  of  a  tooth  on  an  11-pitch  gear. 

18.  Find  the  pitch  of  a  75-tooth  gear  having  an  outside 
diameter  of  19J". 

19.  The  pitch  diameter  of  a  gear  is  1.8182"  and  the  pitch 
is  22.    Find  the  number  of  teeth  and  the  outside  diameter. 

20.  The  circular  pitch  of  a  gear  is  If".    Find  the  working 
depth  of  a  tooth. 


CENTER  DISTANCE 


91 


Cutting  Spur  Gears.  In  cutting  spur  gears  on  a  milling 
machine  a  cutter  of  the  proper  pitch  is  placed  on  the  arbor 
of  the  machine,  the  table  of  which  is  set  at  zero.  The  center 
line  on  the  cutter  teeth  must  be  central  with  the  axis  of  the 
gear  blank,  which  is  held  on 
an  arbor  between  the  dividing- 
head  and  tailstock  centers.  The 
gear  is  indexed  by  one  of  the 
methods  described  in  Chapter  V. 

Center  Distance.  In  speaking 
of  a  pair  of  gears  the  smaller 
gear,  if  one  is  smaller  than  the 
other,  is  called  the  pinion,  the 
other  being  called  the  gear.  In 
finding  the  center  distance  Dc 
of  two  spur  gears  in  mesh,  we 
use  the  formula 


CUTTING    A    SPUR    GEAR   ON    A 
MILLING   MACHINE 


in  which  Ng  and  Np  represent 

the  number  of  teeth  on  the  gear  and  the  number  of  teeth 

on  the  pinion  respectively,  and  Pd  is  the  diametral  pitch. 

There  is  also  a  convenient  rule  used  in  finding  the  pitch 
diameters  of  two  spur  gears  in  mesh,  given  the  center  distance 
and  the  ratio  vg :  vp  of  the  velocities  of  the  gear  and  the  pinion 
in  lowest  terms.  This  rule  expressed  by  formulas  is  as  fol- 
lows, Dpg  and  Dpp  being  the  pitch  diameters  of  the  gear  and 
the  pinion  respectively: 

Dpg  =  2vhx 


92  GEARS 

Illustrative  Problems.  1.  Two  gears  in  mesh  have  90  teeth 
and  70  teeth  respectively.  If  the  pitch  of  the  gears  is  10, 
what  is  the  center  distance? 

90  +  70  =  160  = 
2  x  10        20 

Therefore  the  center  distance  of  the  pair  of  gears  is  8". 

2.  Find  the  pitch  diameters  of  two  gears  that  are  12" 
between  centers  and  have  velocities  hi  the  ratio  of  1  to  2. 

The  smaller  gear,  or  pinion,  will  obviously  have  the  larger  velocity, 
so  that  in  the  ratio  of  velocities  2  represents  the  velocity  of  the  pinion 
and  1  represents  that  of  the  gear. 

D      =  2  x  2  x  -I?-  =  2  x  2  x  4  =  16. 


Therefore  the  pitch  diameters  of  the  gear  and  the  pinion  are  16" 
and  8"  respectively. 

Exercises.    Spur  Gears 

1.  Two  gears  in  mesh  have  36  teeth  and  24  teeth  respec- 
tively.   If  their  pitch  is  8,  what  is  the  center  distance  ? 

2.  Two  gears  in  mesh  have  128  teeth  and  76  teeth  respec- 
tively.   If  their  pitch  is  18,  what  is  the  center  distance  ? 

3.  If  the  ratio  of  the  velocities  of  two  gears  in  mesh  is 
4  to  5  and  the  center  distance  of  the  gears  is  27",  what 
are  the  pitch  diameters? 

4.  If  the  center  distance   of  two  gears  in    mesh  is   16" 
and  the  ratio  of  the  velocities  is  2  to  3,  what  are  the  pitch 
diameters  of  the  gears  ? 

5.  The   ratio   of  the  velocities   of  two  gears  in  mesh  is 
3  to  4,  and  the  center  distance  is  21".    If  the  gears  have  a 
pitch  of  10,  what  is  the  outside  diameter  of  each  gear? 


SPUE  GEAKS  93 

6.  In  Ex.  5  find  the  number  of  teeth  on  each  of  the  gears. 

7.  If   two    6-pitch   gears  in   mesh   have  132   teeth  and 
66  teeth  respectively,  what  is  the  center  distance  ? 

8.  If  the  velocity  ratio  of  two  gears  in  mesh  is  5  to  6 
and  the  distance  between  the  centers  is  33",  what  are  the 
pitch  diameters  of  the  gears  ? 

9.  If  the  pitch  in  Ex.  8  is  6,  find  the  outside  diameters. 

10.  Find  the  number  of  teeth  on  each  of  the  gears  in  Ex.  8. 

11.  If  two  3-pitch  gears  in  mesh  have  182  teeth  and  96  teeth 
respectively,  what  is  the  center  distance  ? 

12.  If  the  velocity  ratio  of  two  gears  in  mesh  is  3  to  1 
and  the  distance  between  the  centers  of  the  gears  is  48", 
what  are  the  pitch  diameters? 

13.  If  the  pitch  in  Ex.  12  is  4,  find  the  outside  diameters. 

14.  Find  the  number  of  teeth  on  each  of  the  gears  in  Ex.  12. 

15.  The  center  distance  of  two  gears  in  mesh  is  6".    One 
gear  has  48  teeth  and  its  pitch  is  16.    Find  the  number  of 
teeth  and  the  outside  diameter  of  the  other  gear. 

16.  If  the  center  distance  of  two  gears  in  mesh  is  12",  and 
the  gears  have  a  velocity  ratio  of  4  to  6,  find  the  pitch  diam- 
eters.   If  the  pitch  of  the  gears  is  10,  find  the  number  of 
teeth  on  each  gear  and  also  find  the  outside  diameters. 

17.  If  the  gears  in  Ex.  16  are  cut  on  a  milling  machine 
which  is  equipped  with  a  Cincinnati  dividing  head,  find  the 
indexing  required  in  cutting  each  gear. 

18.  If  the  pitch  diameters  of  two  gears  in  mesh  are  41" 
and  13J"  respectively,  find  the  velocity  ratio  and  the  dis- 
tance between  the  centers  of  the  gears. 

19.  Two  gears  mesh  together,  one  having  76  teeth  and  an 
outside  diameter  of  5",  and  the  other  49  teeth  and  an  outside 
diameter  of  3".    Find  the  center  distance  of  the  gears. 


94  GEARS 

Comparative  Sizes  of  Gear  Teeth.  In  order  that  the  student 
may  have  an  idea  of  the  comparative  sizes  of  the  gear  teeth 
that  he  is  using,  several  figures  are  shown  on  page  95.  In 
some  of  these  cases  both  Pd  and  Pc  are  given,  but  either  may 
be  found  from  the  other  by  the  formulas  on  page  88. 

Exercises.    Gear  Teeth 

Find  the  circular  pitch  of  a  gear  when  the  diametral  pitch  in 
each  case  is  as  follows : 

1.  f.          3.  14.        5.  18.         7.  22.         9.  28.         11.  38. 

2.  1J.        4.  16.        6.  20.        8.  24.       10.  32.         12.  46. 

Find  the  diametral  pitch  of  a  gear  when  the  circular  pitch  in 
each  case  is  as  follows : 

13.  0.0628".       15.  0.0873".       17.  0.1047".      19.  0.2856". 

14.  0.0714".       16.  0.0924".       18.  0.1208".      20.  0.8976". 

21.  Find   the  thickness  on  the  pitch  line  of  a  \-Pd  gear 
tooth;  of  a  1|-^J  gear  tooth. 

A  \-Pd  gear  tooth  is  a  tooth  on  a  gear  which  has  a  pitch  of  \. 

22.  Find  the  addendum  of  a  4-inch-^  gear  tooth. 

A  4-inch-Pc  gear  tooth  is  a  tooth  on  a  gear  in  which  Pc  =  4". 

23.  Find  the  whole  depth  of  a  %\-Pd  gear  tooth. 

24.  Find  the  thickness  on  the  pitch  line  of  a  i|-inch-JJ 
gear  tooth;  of  a  lT3g-inch-P  gear  tooth. 

25.  If  the  pitch  diameter  of  a  spur  gear  is  2.8571"  and 
there  are  40  teeth,  what  is  the  pitch  of  the  gear  ? 

26.  Find  the  addendum  of  a  1J--ZJ  gear  tooth. 

27.  Find  the  clearance  of  a  Jg-inch-p;  gear  tooth. 

28.  Find  the  whole  depth  of  a  38-P,  gear  tooth. 

29.  Find  the  working  depth  of  a  |-inch-^  gear  tooth. 


SIZES  OF  GEAR  TEETH 


95 


36  Pd  32  Pd  28  Pd  24  Pd 


20  P 


16  P 


14  P 


2  Pd     Pc  =  0.2618"  10 /Jd      />c  =  0.3142''  9  Pd  7% -0.3491' 


3  Pd  Pc  =  1.0472" 

COMPARATIVE    SIZES    OF    GEAR   TEETH 


96  GEAKS 

Exercises.    Review 

1.  Find  the  number  of  teeth  on  a  spur  gear  having  an 
outside  diameter  of  4.4"  and  a  pitch  of  10. 

2.  Find  the  pitch  diameter  of  a  spur  gear  that  has  32 
teeth  and  a  pitch  of  10. 

3.  In  Ex.  2  find  the  outside  diameter. 

4.  Find  the  pitch  of  a  gear  in  which  ^  =  6.2832". 

5.  Find  the  center  distance  of  two  12-JJ  gears  in  mesh 
that  have  66  teeth  and  54  teeth  respectively. 

6.  Find  the  circular  pitch  of  a  gear  that  has  a  pitch  of  34. 

7.  Find  the  whole  depth  of  a  tooth  on  a  rack  that  has  a 
linear  pitch  of  0.1745". 

When  referring  to  a  rack  the  term  **  linear  pitch  "  is  used  in  place 
of  the  term  "  circular  pitch." 

8.  Find  the  pitch  diameter  of  a  gear  with  42  teeth  and  a 
circular  pitch  of  0.2618". 

9.  The  velocity  ratio  of  two  gears  in  mesh  is  3  to  5  and 
the  center  distance  is  24".    Find  the  pitch  diameters. 

10.  If  the  pitch  of  the  gears  in  Ex.  9  is  4,  what  is  the 
number  of  teeth  on  each  gear? 

11.  In  Ex.  10  what  is  the  outside  diameter  of  each  gear? 

12.  Find  the  pitch  of  a  spur  gear  that  has  80  teeth  and 
an  outside  diameter  of  16.4". 

13.  If  a  spur  gear  has  a  pitch  of  36,  what  is  the  working 
depth  of  a  tooth? 

14.  If  the  circular  pitch  of  a  gear  is  0.1653",  what  is  the 
addendum  ?   the  thickness  of  tooth  on  the  pitch  line  ? 

15.  If  a  gear  has  140  teeth  and   an  outside  diameter  of 
10}",  what  is  the  pitch  ? 


BEVEL  GEARS 


97 


BEVEL    GEARS 


Bevel  Gears.  Bevel  gears  are  used  for  transmitting  rotary 
motion  from  one  shaft  to  another  where  the  two  shafts  are 
not  parallel  to  each  other. 

While  the  two  shafts  are  generally  at 
right  angles  to  each  other,  this  is  not  essen- 
tial;  gears  can  be  made  to  allow  for  any 
particular  angle  desired. 

As  before,  one  of  the  two  gears,  the 
smaller,  if  there  is  a  smaller,  is  called 
the  pinion,  the  other  one  being  then 
called  the  gear. 

The  teeth  of  bevel  gears  are  con- 
structed on  imaginary  friction  cones  in  the  same  way  that 
the  teeth  of  spur  gears  are  constructed  on  imaginary  fric- 
tion cylinders. 

Such  terms  as  diametral  pitch,  circular  pitch,  whole  depth,  pitch  diameter, 
and  outside  diameter  are  used  with  the  same  general  meaning  as  in  the 
case  of  spur  gears. 

Since  the  teeth  of  a  bevel  gear  constantly  become  thinner 
as  they  approach  the  apex  of  the  cone,  in  cutting  bevel 
gears  with  a  rotary  cutter  on  a  milling  machine,  it  is  not 
possible  to  obtain  perfect  finished  teeth.  The  teeth  are  cut 
to  approximate  size  by  offsetting  the  cutter  and  rotating 
the  blank,  and  the  final  thinning  of  the  small  end  of  the 
teeth  is  generally  done  by  filing.  At  the  large  end  of  the 
teeth  on  a  bevel  gear  the  circular  pitch,  the  thickness  of  tooth 
on  the  pitch  line,  and  the  whole  depth  of  tooth  are  identical 
with  those  of  a  spur  gear  of  the  same  size  and  pitch. 

Bevel  gears  are  always  figured  in  pairs,  and  we  shall  con- 
sider only  such  cases  as  those  in  which  the  shafts  are  at  right 
angles  to  each  other. 

The  method  of  computing  the  essential  measurements  for  cutting  a 
pair  of  bevel  gears  is  shown  on  page  101. 


98 


GEARS 


Terms  Used.  In  addition  to  the  terms  used  with  spur 
gearing  given  on  page  86,  the  following  terms,  illustrated  in 
the  figure  below,  are  used  in  connection  with  bevel  gears : 

Addendum  angle.  The  angle  (V)  between  the  pitch  line 
and  the  top  line  of  a  tooth  extended  to  the  vertex. 


TERMS  USED  WITH  BEVEL  GEARS 

a,  addendum ;  a',  addendum  angle ;  (7,  cutting  angle ;  d,  dedendum ;  d',  deden- 

dum  angle ;  D0,  outside  diameter ;  Dp,  pitch  diameter;  F,  width  of  face ;  F',  face 

angle ;  K,  pitch-cone  angle ;  P,  pitch  line ;  /?,  pitch-cone  radius ;   W,  whole  depth ; 

x,  angular  addendum ;  Z,  vertex 

Angular  addendum.  The  straight-line  distance  (x)  from 
the  pitch  circle  to  the  outside  circle. 

Twice  the  angular  addendum  is  called  the  diameter  increment. 

Cutting  angle.  The  angle  (C)  between  the  bottom  of  a 
groove  and  the  axis  of  the  gear. 

This  angle  is  the  same  as  the  angle  at  which  the  dividing  head  is 
set  while  cutting  the  teeth. 

Dedendum  angle.  The  angle  (d1)  formed  by  the  bottom 
of  a  groove  and  the  pitch  line. 


TERMS  USED 


99 


Face  angle.  The  angle  (/'7')  with  the  axis  to  which  the 
gear  blank  is  turned. 

Pitch  circle.  The  circle  formed  by  a  cross  section  of  the 
imaginary  friction  cone,  the  diameter  of  the  circle  being  the 
pitch  diameter. 

Since  bevel  gears  are  constructed  on  cones  they  may  have  several 
pitch  circles,  as,  for  example,  a  pitch  circle  at  the  larger  end  of  the  teeth 
and  one  at  the  smaller  end.  In 
speaking  of  the  pitch  of  a  bevel 
gear  we  always  mean  the  number  of 
teeth  per  unit  of  the  diameter  of  the 
largest  pitch  circle ;  that  is,  the  diam- 
eter Dp  in  the  figure  on  page  98. 

Pitch-cone  angle.  The  angle 
(7f)  with  the  axis  of  the  imag- 
inary friction  cone. 

Pitch-cone  radius.  The  radial 
distance  (J£)  from  the  vertex 
to  the  largest  pitch  circle. 

This  distance  is  measured  from 
the  vertex  Z  to  the  circle  of  which 
Dp  is  the  diameter  in  the  figure  on 
page  98.  If  the  surface  of  the  imag- 
inary friction  cone  were  laid  out  flat  it  would  be  a  sector  of  a  circle. 
The  pitch-cone  radius  is  the  radius  of  this  circle,  and  is  sometimes 
called  the  apex  distance,  or  vertex  distance. 

Pitch  line.  A  line  equivalent  to  the  slant  height  of  the 
imaginary  friction  cone ;  more  specifically,  that  part  (P)  of 
the  line  lying  within  the  teeth.  Also,  as  in  spur  gearing, 
the  circumference  of  the  greatest  pitch  circle ;  that  is,  the 
circle  measured  by  Dp  in  the  figure  on  page  98. 

Vertex.   The  apex  (Z)  of  the  cone  on  which  the  gear  is  cut. 

Width  of  face.  The  length  (.F)  of  the  teeth,  usually  cut 
so  as  to  be  about  one  fourth  the  pitch-cone  radius. 


CUTTING    A    BEVEL    GEAR    ON    A 
MILLING    MACHINE 


100  GEARS 

Symbols.  In  the  formulas  and  in  computations  relating  to 
bevel  gears  we  shall  use  the  following  symbols : 

a  =  addendum  F1  =  face  angle 

a'  =  addendum  angle  K  =  pitch-cone  angle 

C  =  cutting  angle  N  =  number  of  teeth 

d  =  dedendum  Pd  =  diametral  pitch 

d'  =  dedendum  angle  R  =  pitch-cone  radius 

D0  =  outside  diameter  W=  whole  depth 

Dp  =  pitch  diameter  x  =  angular  addendum 

The  subscript  g  for  "  gear  "  or  p  for  "  pinion  "  may  be  added  to  indi- 
cate that  a  certain  symbol  applies  to  the  gear  or  to  the  pinion.  Thus  Cg 
is  the  cutting  angle  of  the  gear  and  Cp  is  that  of  the  pinion. 

Bevel-Gear  Formulas.  The  following  formulas  may  be 
referred  to  in  solving  the  exercises  on  bevel  gears : 


All  the  formulas  in  which  the  subscripts  g  for  "  gear  "  and  p  for 
"  pinion  "  are  not  used,  apply  to  both  the  gear  and  the  pinion.  We  may 
indicate  that  a  formula  applies  to  the  gear  or  to  the  pinion  by  adding 
the  proper  subscripts,  as  in  the  following  cases : 

D»=j?         D»  =  T? 

^d  rd 

The  result  found  from  the  formula  for  d  is  actually  the  dedendum 
plus  the  clearance,  but  in  bevel  gears  we  treat  this  as  the  dedendum. 

If  the  machinist  uses  formed  cutters  and  the  correct  indexing,  the 
thickness  on  the  pitch  line  at  the  large  end  of  a  tooth  will  be  correct.  This 
measurement  is  usually  checked  with  a  table  of  chordal  thicknesses. 


BEVEL-GEAR  FORMULAS 


101 


Illustrative  Problem.    Make  the  calculations  for  the  pair  of 
-7J  bevel  gears  shown  below  in  which  Ng  =  48  and  Np  =  24. 


(1)  tan  Kg  =     a.  =        =  2.0000,  and  hence  Kg  =  63°  26'. 
JNp      24 

Kp  =  90°  -  Kg  =  90°  -63°  26'  =  26°  34'. 


Hence    Dp    of    gear    is 
4"  and  Z)p  of  pinion 


is  2". 


=  4 


•>  v  0  8Q44 


Hence  D0  of  gear  is  4.075"  and  D0  of  pinion  is  2.149". 

Dp  4 

~  2  sin  Ky  ~  2  x  0.8945  5  ' 

Hence  R  for  both  gear  and  pinion  is  2.236". 


=  0.1798.  Hence  Wg  and  Wp  are  each  0.180". 


(5)  W  =  = 


(6)  a  =       =       =  0.0833.   Hence  ag  and  ap  are  each  0.083". 


(7)  d  = 


=  0.0964.   Hence  dg  and  dp  are  each  0.096". 


(8)  tan  a'  =  -     =    '          =  0.0373,  and  a'  (both  gear  and  pinion)  =  2°  8'. 

''' 


(9)  tan  d'  =  —  =    '          =  0.0431,  and  ef  (both  gear  and  pinion)  =  2°  287. 
-R      2.23o9 

(10)  Cg  =  Kg  -d'  =  63°  26'  -  2°  28'  =  60°  58'. 
Cp  =  Kp-d'  =  26°  34'  -  2°  28'  =  24°  6'. 

(11)  F'g  =  Kg  +  a'  =  63°  26'  +  2°  8'  =  65°34X. 
F'  =  K   +  a'  =  26°34'  +  2°8/  =  28°42/. 


102  GEAKS 

Exercises.    Bevel  Gears 

For  a  pair  of  level  gears,  if  it  is  required  that  Pd  —  9,  Ng  =  36, 
and  Np  =  18,  find  each  of  the  following  : 

1.  The  pitch  diameters  of  both  the  gear  and  the  pinion. 

2.  The  pitch-cone  angles  of  both  the  gear  and  the  pinion. 

3.  The  outside  diameters  of  both  the  gear  and  the  pinion. 

4.  The  pitch-cone  radius. 

5.  The  whole  depth  of  the  teeth. 

6.  The  addendum  and  the  dedendum. 

7.  The  addendum  angle  and  the  dedendum  angle. 

8.  The  cutting  angles  of  both  the  gear  and  the  pinion. 

9.  The  face  angles  of  both  the  gear  and  the  pinion. 

10.  The  thickness  of  tooth  on  the  pitch  line. 

This  measurement  at  the  large  end  of  the  teeth  may  be  calculated 
from  one  of  the  spur-gearing  formulas  for  t  given  on  page  88,  but  it 
is  not  an  essential  measurement  when  the  gear  is  cut  with  formed 
cutters.  As  a  check,  however,  the  thickness  of  tooth  is  usually  measured 
with  a  special  gear-tooth  vernier  caliper  and  compared  with  a  table  of 
chordal  thicknesses  of  gear  teeth,  since  the  caliper  measures  the  chordal 
thickness  and  not  the  thickness  of  tooth  along  the  pitch  circle. 

Make  all  the  necessary  calculations  for  cutting  each  of  the 
following  pairs  of  bevel  gears : 

11.  Gear,  52  teeth;  pinion,  36  teeth;  pitch,  14. 

12.  Gear,  45  teeth ;  pinion,  15  teeth ;  pitch,  10. 

13.  Gear,  56  teeth ;  pinion,  32  teeth ;  pitch,  8. 

14.  Gear,  48  teeth;  pinion,  12  teeth;  pitch,  14. 

15.  Gear,  60  teeth;  pinion,  18  teeth;  pitch,  12. 

The  elements  required  in  making  the  calculations  for  cutting  a  pair 
of  bevel  gears  are  shown  on  page  101. 


SPIRAL  GEARS 


103 


SPIRAL   GEARS 


Spiral  Gears.  Gears  in  which  the  teeth  are  wound  spirally 
around  the  axis  instead  of  being  cut  parallel  to  the  axis, 
as  in  a  spur  gear,  are  called  spiral  gears  or  helical  gears. 

Although  it  is  the  common  term,  the  name 
"  spiral  gear  "  is  not,  strictly  speaking,  correct, 
since  the  teeth  are  not  wound  around  the  axis 
in  a  spiral,  but  in  a  helix. 

Spiral  gears  are  essentially  cylinders 
with  a  succession  of  equally  spaced 
spiral  grooves  cut  on  their  periphery, 
and  hence  are  more  nearly  like  spur 
gears  than  bevel  gears.  Spiral  gears 
may  have  their  axes  parallel,  as  in  the 
case  of  spur  gears,  or  the  axes  may  be 
at  any  desired  angle  to  each  other. 

A  tooth  of  a  spiral  gear  is  much  like 

the  thread  of  a  screw,  although  it  does  not  have  the  same 
cross  section  nor  is  it  meant  to  do  the  same  kind  of  work. 

In  this  connection  it  should  be  remembered  that  the  meaning  of  the 
term  "  pitch  "  when  applied  to  gears  is  different  from  that  when  the  term 
is  applied  to  screw  threads. 

In  the  treatment  of  spiral  gears  we  shall  consider  only  those  cases 
in  which  a  variation  in  the  center  distance  of  the  gears  is  permissible ; 
that  is,  those  in  which  the  center  distance  is  said  to  be  approximate. 

Uses  of  Spiral  Gears.  Spiral  gears  are  used  when  high 
speed  with  smooth  and  noiseless  action  is  desired.  The 
absence  of  noise  is  due  to  the  fact  that  the  engagement  of 
the  teeth  is  gradual,  for  instead  of  striking  a  full  line  of 
contact  at  once,  the  teeth  roll  on  each  other  with  a  sliding 
motion,  one  set  engaging  while  the  preceding  set  is  still  in 
contact.  Thus,  the  smoothness  of  action  of  spiral  gears  is 
hardly  impaired  by  wear,  and  these  gears  are  used  extensively 
in  automobiles  and  many  different  types  of  machines. 


104 


GEARS 


Terms  Used.  In  addition  to  the  terms  used  in  connection 
with  spur  gearing,  as  given  on  page  86,  the  following  special 
terms  are  used  with  reference  to  spiral  gears : 

Axial  pitch.  The  distance  from  the  center  of  one  tooth  to 
the  center  of  the  next  tooth,  measured  parallel  to  the  axis 
of  the  gear. 

Since  it  is  made  parallel  to  the  axis,  this  meas- 
urement is  similar  to  the  pitch  of  a  screw  thread. 

Circular  pitch.  The  distance  from  the 
center  of  one  tooth  to  the  center  of  the 
next  tooth,  measured  on  the  pitch  circle. 

This  measurement  is  made  at  right  angles  to 
the  axis  of  the  gear  as  in  spur  gearing.  On 
account  of  the  different  uses  of  the  word  "  pitch  " 
with  spiral  gears,  this  is  sometimes  called  the  SPIRAL-GEAR  TEETH 
real  circular  pitch,  and  the  diametral  pitch  corre-  This  shows  the  differ- 
sponding  to  this  is  called  the  real  pitch.  ence  between  the  ways 

in  which  the   normal 

Lead.    The  distance  that  the  spiral  on    circular  pitch  (N)  and 

,  .  ,       .,  .,  ,  .  the  axial  pitch  (A)  are 

which   the   teeth   are   cut   advances   in    a  measured 

single  turn. 

Normal  circular  pitch.  The  distance  from  the  center  of  one 
tooth  to  the  center  of  the  next  tooth,  measured  at  right 
angles  to  the  center  line  of  the  teeth. 

This  distance  is  shorter  than  the  real  circular  pitch.  The  section  of 
the  teeth  which  shows  the  normal  circular  pitch  is  called  the  true  section. 

Normal  diametral  pitch.  The  diametral  pitch  corresponding 
to  the  normal  circular  pitch. 

The  diametral  pitch  of  the  cutter  used  in  cutting  the  teeth  is  the 
same  as  the  normal  pitch  (normal  diametral  pitch)  of  the  gear. 

Tooth  angle.  The  angle  measured  at  the  pitch  circle  between 
the  center  line  of  the  teeth  and  a  line  parallel  to  the  axis. 

This  angle  is  also  called  the  spiral  angle  of  the  gear. 


SPIRAL  GEARS 


105 


Cutting  Spiral  Gears.  The  teeth  of  spiral  gears  when  cut 
on  a  milling  machine  are  generally  formed  with  ordinary 
involute  spur-gear  cutters,  the  cutter  of  the  proper  pitch 
and  number  for  the  given  spiral  gear  being  selected  as 
described  on  page  107. 

The  gear  cutter  is  fastened  firmly  on  the  arbor  of  the 
milling  machine.  The  dividing  head  and  tailstock  are  set 
in  position  on  the  table 
of  the  machine  with  their 
centers  in  alignment  and 
central  with  the  cutter. 

The  change  gears  needed 
to  give  the  required  spiral 
lead  are  placed  in  their 
proper  positions,  as  shown 
on  page  74,  and  the  divid- 
ing head  is  set  to  index 
the  number  of  teeth  to  be 
cut  on  the  gear. 

The  gear  blank  is  pressed 
firmly  on  the  mandrel  and 
is  mounted  between  the 
dividing-head  and  the  tail- 
stock  centers.  The  blank 

is  brought  under  the  cutter,  and  the  table  is  set  to  the 
required  tooth  angle  and  clamped.  The  blank  is  then  moved 
clear  of  the  cutter,  the  knee  is  raised  for  the  required  depth 
of  cut,  and  the  vertical  dial  is  set  at  zero.  The  machine  is 
then  started  and  the  table  moved  by  hand  until  the  gear 
blank  touches  the  cutter,  when  the  power  feed  is  thrown  in. 
In  returning  the  blank  for  another  cut,  the  knee  should  be 
lowered  until  the  blank  clears  the  cutter,  thus  preventing 
the  cutter  from  dragging  through  the  cut  just  made. 


CUTTING   A    SPIRAL    GEAR    ON    A 
MILLING   MACHINE 


106  GEARS 

Symbols.  The  following  symbols  are  used  in  the  formulas 
and  computations  for  spiral  gears: 

A  =  tooth  angle  Pn  =  normal  pitch 

Dc  =  center  distance  r  =  ratio  of  gear  to  pinion 

D0  =  outside  diameter  S  =  angle  of  shafts 

Dp  =  pitch  diameter  Tc  =  number    for    selecting 

L  =  lead  of  spiral  gear  cutter 

N  =  number  of  teeth  W  =  whole  depth 

In  order  to  distinguish  between  the  symbols  for  the  gear  and  those 
for  the  pinion,  we  add  the  subscript  g  or  p  as  in  previous  cases.  Thus, 
Ag  is  the  tooth  angle  of  the  gear  and  Ap  is  the  tooth  angle  of  the  pinion. 

The  ratio  r  of  gear  to  pinion  is  the  ratio  of  the  number  of  teeth  on 
the  gear  to  the  number  on  the  pinion.  If  the  numbers  are  the  same, 
the  ratio  is  commonly  said  to  be  equal. 

Spiral-Gear  Formulas.  The  following  formulas  are  used 
with  spiral  gears  when  the  ratio  is  unequal,  the  center  dis- 
tance is  approximate,  and  the  shafts  are  at  any  angle  : 


When  the  subscripts  g  for  "gear"  and/>  for  "pinion"  are  not  used, 
the  formulas  apply  to  both  gear  and  pinion,  and  in  distinguishing 
between  the  formulas  for  the  gear  and  those  for  the  pinion  the  proper 
subscripts  may  be  added,  as  in  the  following  cases  : 

Lg  =  'Tfl)pg  cot  A  g  Lp  =  irDpp  cot  Ap 

In  the  above  formula  for  Npt  if  Ag  =  Ap,  we  have 
,T  _2DcPncosA 


SPIRAL  GEARS 


107 


Gear  Cutters.  In  the  Brown  and  Sharpe  system  of  cutters 
for  spur  gears  with  involute  teeth,  eight  different  shapes  of 
cutters,  as  shown  in  the  table  below,  are 
made  for  each  pitch.  The  cutters  are  num- 
bered according  to  the  number  of  teeth  to 
be  cut  on  the  gear,  a  #8  %-Pd  cutter,  for 
example,  being  required  for  cutting  8-Pz  spur 
gears  with  12  or  13  teeth.  For  cutting  a 
spiral  gear  the  proper  number  of  cutter  of 
the  normal  pitch  of  the  gear  is  selected 
from  the  table  according  to  the  number  found  INVOLUTE  GEAR 
by  the  formula  for  Tc  on  page  106.  CUTTER 

If  with  these  cutters  the  correct  indexing  and  the  proper  depth  of 
cut  are  used,  the  thickness  of  tooth  on  the  pitch  line  will  be  correct. 


NUMBER  OF 
CUTTER 

NUMBER  OF 
TEETH 

NUMBER  OF 
CUTTER 

NUMBER  OF 
TEETH 

1 

135  to  rack 

5 

21  to  25 

2 

55  to  134 

6 

17  to  20 

3 

35  to  54 

7 

14  to  16 

4 

26  to  34 

8 

12  to  13 

A  #1  cutter  is  used  in  cutting  gears  with  135  or  more  teeth  or  in 
cutting  the  teeth  on  racks. 

Change  Gears.  The  change  gears  required  to  cut  the 
proper  lead  on  a  spiral  gear  may  be  found  by  the  method 
described  on  page  75.  Since  the  spiral  leads  as  calculated 
from  the  formula  on  page  106  usually  contain  decimals  which 
are  difficult  to  factor,  it  is  customary,  in  practice,  to  use 
tables  of  spiral  leads,  which  show  the  leads  obtainable  with 
every  possible  arrangement  of  the  change  gears  furnished 
with  the  dividing  head.  When  an  exact  lead  cannot  be  found 
in  the  table  the  nearest  obtainable  lead  is  generally  used. 


108  GEARS 

Illustrative  Problem.  Make  the  calculations  for  cutting  a 
pair  of  spiral  gears  in  which  it  is  required  that  r  =  4,  Pn  =  6, 
Dc  (approximate)  =  12",  Ag  =  30°,  Ap  =  20°,  and  S=  50°. 

r       2DcPncosAffcosAp  _  2  x  12  x  6  x  0.8660  x  0.9397  _  25  o 
p  =     rcoaAp  +  coaAff  4  x  0.9397  +  0.8660 

Such  a  result  is  taken  to  the  nearest  whole  number,  in  this  case  25. 

Ng  =  rNp  =  4  x  25  =  100. 

The  gear  and  pinion  have  100  teeth  and  25  teeth  respectively. 

no  9456 
= 


p      6  x  0.9397 
The  pitch  diameters  of  gear  and  pinion  are  19.246"  and  4.434". 

(3)  Dog  =  Dpg  +  ^  =  19.2456  +  |  =  19.5789. 

•*   n 

DoP  =  Dpp  +  £-  =  4.4340  +  |  =  4.7673. 

•*•  n 

The  outside  diameters  of  gear  and  pinion  are  19.579"  and  4.767". 
•(4)  Lg  =  irDpgcotAg  =  3.1416  x  19.2456  x  1.7321  =  104.7262. 
Lp  =  irDpp  coiAp  =  3.1416  x  4.4340  x  2.7475  =  38.2724. 
The  leads  of  the  spirals  on  gear  and  pinion  are  104.73"  and  38.27". 

1QO     _  154  n 

~ 

301 
cos3Ap      0.93973  "  0.8298  " 

From  the  table  on  page  107  we  see  that  a  #1  Q-P(l  cutter  is  required 
for  the  gear  and  a  #4  6-Prf  cutter  for  the  pinion. 


The  whole  depth  of  tooth  on  both  gear  and  pinion  is  0.360''. 
(7)  Dc  =  J  (Dpg  +  Dpp)  =  $  (19.2456  +  4.4340)  -  11.8398. 
The  "exact"  center  distance  of  the  gears  is  11.840". 


SPIRAL  GEARS  109 

Exercises.    Spiral  Gears  at  Any  Angle 

For  a  pair  of  spiral  gears,  if  it  is  required  that  r  =  2,  Pn  =  10, 
Dc (approximate)  =  10",  Ag=30°,  Ap  =  20°,  and  S=  50°,  find 
each  of  the  following : 

1.  The  number  of  teeth  on  the  gear  and  on  the  pinion. 

2.  The  pitch  diameters  of  both  the  gear  and  the  pinion. 

3.  The  outside  diameters  of  both  the  gear  and  the  pinion. 

4.  The  leads  of  the  spirals  on  both  the  gear  and  the  pinion. 

5.  The  numbers  of  the  two  cutters  required. 

6.  The  whole  depth  of  the  teeth. 

7.  The  exact  center  distance  of  the  gears. 

For  a  pair  of  spiral  gears,  if  it  is  required  that  r  =  3,  J^=4, 
Dc  (approximate)  =  16' f,  Ag=40°,  Ap=20°,  and  S  =  60°,  find 
each  of  the  following : 

8.  The  number  of  teeth  on  the  gear  and  on  the  pinion. 

9.  The  pitch  diameters  of  both  the  gear  and  the  pinion. 

10.  The  outside  diameters  of  both  the  gear  and  the  pinion. 

11.  The  leads  of  the  spirals  on  both  the  gear  and  the  pinion. 

12.  The  numbers  of  the  two  cutters  required. 

13.  The  whole  depth  of  the  teeth. 

14.  The  exact  center  distance  of  the  gears. 

15.  Make  the  necessary  calculations  for  cutting  a  pair  of 
spiral  gears,  it  being  required  that   Dc  (approximate)  =  1", 
r  =  l^,  Pn=S,  Ag=25°,  4,  =  15°,  and  S=40°. 

16.  Make  the  necessary  calculations  for  cutting  a  pair  of 
spiral  gears,  it  being  required  that  Dc  (approximate)  =  1 2", 
r  =  4,  %=$,  4,  =  30°,  4,  =  30°,  and  S=60°. 

In  finding  Np  use  the  formula  given  at  the  foot  of  page  106. 


110 


GEARS 


Spiral  Gears  with  Parallel  Shafts.  When  the  ratio  of  a 
pair  of  spiral  gears  is  unequal,  the  center  distance  approxi- 
mate, and  the  shafts  parallel,  as  shown  in  the  illustration, 
the  tooth  angle  of  the  gear  is  equal  to  the  tooth  angle  of  the 
pinion  ;  that  is,  Ag  =  Ap. 

For  example,  make  the  calculations  for  cutting  a  pair  of 
spiral  gears  with  parallel  shafts,   if   it 
is  required  that  Dc  (approximate)  =  10", 
Ng=  48,  Np=  24,  %  =  4,  and  A=  24°. 

CD  ft,         "• 


99      PncosA 


4  x  0.9135 
24 


=  13.1363. 


=  6.5681. 


4  x  0.9135 

The  pitch  diameters  of  gear  and  pinion 
are  13.136"  and  6.568"  respectively. 


(2)  Dog  = 


=  13.1363  +  f  =  13.6363. 


—  =  6.5681 

* 


=  7.0681. 


SPIRAL    GEARS    WITH 
PARALLEL    SHAFTS 


The  outside  diameters  of  gear  and  pinion  are  13.636"  and  7.068". 
(3)  Lg  =  TrDpgcotA  =  3.1416  x  13.1363  x  2.2460  =  92.6902. 
Lp  =  irDppcotA  =  3.1416  x  6.5681  x  2.2460  =  46.3446. 
The  leads  of  the  spirals  on  gear  and  pinion  are  92.69"  and  46.34". 

Ng  48  48 

cos3.!  ~ 

Nn 


(4)  Tcg  = 


TCP 


0.91353      0.762.3 
24  24 


=  63.0. 


=  31.5. 


ooeM      0.91353      0.7623 
From  the  table  on  page  107  we  see  that  a  #2  4-Pd  cutter  is 
required  for  the  gear  and  a  #4  4-Pd  cutter  for  the  pinion. 


(5)  W  = 


-*    » 


=  0.5393. 


The  whole  depth  of  tooth  on  both  gear  and  pinion  is  0.539". 
(6)  Dc  =  $  (Dpg  +  Dpp)  =  $  (13.1363  +  6.5681)  =  9.8522. 
The  "exact"  center  distance  of  the  gears  is  9.852". 


SPIRAL  GEARS  111 

Exercises.    Spiral  Gears  with  Parallel  Shafts 

1.  For  a  certain  pair  of  spiral  gears  it  is  required  that  the 
shafts  be  parallel  at  an  approximate  center  distance  of  2^", 
that  JV^=32,  Np=20,  Pn=l2,  and  ^=18°.    Find  the  pitch 
diameters  of  both  the  gear  and  the  pinion. 

2.  In  Ex.  1  find  the  outside  diameters  of  both  the  gear 
and  the  pinion. 

3.  In  Ex.  1  find  the  leads  of  the  spirals  on  both  the  gear 
and  the  pinion. 

4.  In  Ex.  1  find  the  number  of  the  cutter  to  be  used  for 
cutting  the  gear  and  the  same  for  the  pinion. 

5.  In  Ex.  1  find  the  whole  depth  of  the  teeth. 

6.  In  Ex.  1  find  the  exact  center  distance. 

For  a  pair  of  spiral  gears,  if  it  is  required  that  the  shafts 
be  parallel,  that  Dc  (approximate)  =  12-j-",  r  =  1-j,  Pn  =  6,  and 
A  =  20°,  find  each  of  the  following : 

7.  The  number  of  teeth  on  the  gear  and  on  the  pinion. 
In  finding  Np  use  the  formula  given  at  the  foot  of  page  108. 

8.  The  pitch  diameters  of  both  the  gear  and  the  pinion. 

9.  The  outside  diameters  of  both  the  gear  and  the  pinion. 

10.  The  leads  of  the  spirals  on  both  the  gear  and  the  pinion. 

11.  The  number  of  the  cutter  to  be  used  for  cutting  the 
gear  and  the  same  for  the  pinion. 

12.  The  whole  depth  of  the  teeth. 

13.  The  exact  center  distance  of  the  gears. 

14.  A  spiral  gear  with  a  pitch  diameter  of  6.2121"  has  a 
spiral  lead  of  72.83".    Find  the  angle  at  which  the  milling- 
machine  table  was  set  in  cutting  the  gear. 


112 


GEAES 


Worm  Gearing.  When  the  difference  between  the  velocities 
of  two  shafts  is  to  be  great,  worm  gearing  is  often  employed. 

In  this  case  the  driver,  which  is  called  a  worm,  is  a  single- 
thread  or  multiple-thread  screw,  the  threads  of  which  mesh 
with  a  special  form  of  spur  gear  known  as  a  worm  wheel. 

Uses  of  Worm  Gearing.  Worm  gearing  is 
generally  used  when  the  load  is  heavy  and 
when  smoothness  of  action  and  a  large 
reduction  in  velocity  are  required  in  trans- 
mitting rotary  motion  from  one  shaft  to 
another.  Usually  the  two  shafts  are  at 
right  angles  to  each  other. 

Terms  Used.  In  general  the  terms  and 
symbols  which  are  used  with  spur  gearing 
are  also  used  in  connection  with  worm 
gearing,  together  with  the  following  addi- 
tions, which  will  be  understood  from  the 
second  figure  on  page  113: 

Face  angle.  The  angle,  usually  arbitrarily  selected,  to  which 
the  ends  of  the  teeth  on  the  worm  wheel  are  trimmed. 

This  angle  varies  from  about  55°  to  65°,  depending  upon  the  kind 
of  worm  used.  The  face  angle  of  the  wheel  shown  on  page  113  is  60°. 

Gashing  angle  (£).  The  angle  for  swiveling  the  table  of  the 
milling  machine  when  cutting  the  teeth  on  the  worm  wheel  to 
correspond  to  the  spiral  angle  of  the  thread  on  the  worm. 

Linear  pitch  (JJ).  This  is  identical  with  the  circular  pitch 
of  the  worm  wheel  and  is  used  in  speaking  of  the  worm. 

Since  the  two  terms  are  identical,  we  use  the  symbol  Pc  for  both. 

Throat  diameter  (!><).  The  diameter  of  the  worm  wheel 
measured  at  the  center  of  the  curved  teeth. 

The  throat  diameter  is  equivalent  to  the  outside  diameter  of  a  spur 
gear  having  the  same  number  of  teeth  and  the  same  pitch. 


WORM   GEARING 


WORM  GEARING 


113 


Worm-Thread  Formulas.    The  following  formulas  are  used 
with  standard  worm  threads  which  have  an  angle  of  29°  : 

a  =  0.3183PC        W-  0.6866PC 


Dp=D0-2a 


Wt  =  0.3354  Pc 


The  formula  for  D0  gives  the 
outside  diameter  for  a  single- 
thread  worm;  for  a  double- 
thread  worm  use  5^,  and  for  a 


triple-thread  worm  use 

The  width  (JF6)  at  the  root 
of  the  point  of  the  tool  used  in 

Worm-Wheel  Formulas.    The 
with  worm  wheels: 


STANDARD    WORM    THREAD 

a,  addendum;   P,  pitch  line;   Pc, 

linear  pitch;    W,   whole   depth    of 

thread ;  Wb,  width  at  root  of  thread ; 

Wt,  width  at  top  of  thread 


of  the  thread  is  the  width 
cutting  the  worm  thread. 

following  formulas  are  used 


tanG=   PC 


TTD 


'pw 


n    _J^  p    _  IT  CROSS    SECTION    OF    WORM 

P~~~p~  **=|Br  AND   WORM   WHEEL 

•*  d  *  c 

Dc,  center  distance ;  D0,  outside 

DQ  =  Dp  -J-  4  a  t  =  \Pc  '    diameter;    Dp,  pitch  diameter; 

Dt,  throat  diameter 

In  these  formulas  Dpw  is  the  pitch  diameter  of  the  worm  and  Dp  is 
the  pitch  diameter  of  the  worm  wheel. 

The  depth  of  tooth  on  the  worm  wheel,  when  figured  for 
gashing  the  teeth  on  a  milling  machine,  is  taken  as  the  whole 
depth  of  the  worm  thread.  This  evidently  makes  no  allow- 
ance for  clearance,  but  when  the  teeth  are  hobbed,  as  ex- 
plained on  page  115,  an  extra  depth  of  tooth  is  given  to  the 
hob  to  make  the  necessary  allowance. 


114  GEARS 

Illustrative  Problem.  Make  the  calculations  for  cutting  a 
worm  and  worm  wheel  with  a  velocity  ratio  of  40  to  1,  the 
worm  to  be  single-threaded  with  a  linear  pitch  of  0.2618". 

_  TT  _  3.1416  _ 
(1)Pd-Fc~£2618- 

The  diametral  pitch  of  the  worm  wheel  is  12. 

(2)  a  =  -L  =  ±  =±0.0833. 

The  addendum  of  the  worm  wheel  is  0.083". 

(3)  Since  the  velocity  ratio  is  40  to  1,  the  wheel  must  have  40  teeth. 

_^_40 
p  ~  ~P~d  ~  12  ~ 
The  pitch  diameter  of  the  worm  wheel  is  3.333". 

(4)  Dt  =  Dp  +  2  a  =  3.3333  +  (2  x  0.0833)  =  3.4999. 
The  throat  diameter  of  the  worm  wheel  is  3.500". 

(5)  D0  =  Dp  +  4  a  =  3.3333  +  (4  x  0.0833)  =  3.6665". 
The  outside  diameter  of  the  worm  wheel  is  3.667". 

(6)  t  =  \PC  =  %  x  0.2618  =  0.1309. 

The  thickness  of  tooth  on  the  pitch  line  of  the  worm  wheel  is  0.131". 

(7)  D0  =  4  Pc  =  4  x  0.2618  =  1.0472. 

The  outside  diameter  of  the  worm  is  1.047". 

(8)  The  addendum  of  the  worm  is  the  same  as  that  of  the  worm  wheel. 
Dp  =  D0-2a  =  1.0472  -  (2  x  0.0833)  =  0.8806. 

The  pitch  diameter  of  the  worm  is  0.881". 

(9)  Dc  =  }  (Dp  +  Dpw)  =  i  (3.3333  +  0.8806)  =  2.1070. 

The  center  distance  of  the  worm  and  worm  wheel  is  2.107". 

(10)  W  =  0.6866  Pc  =  0.6866  x  0.2618  =  0.1798. 
The  whole  depth  of  the  worm  thread  is  0.180". 

(11)  Wb  =  0.3095  Pc  =  0.3095  x  0.2618  =  0.0810. 

The  width  of  the  point  of  the  worm-thread  tool  is  0.081". 

-p  A  Oftl  Q 

(12)  tan  G  =  ^-  =  3  1416  x  0.8806  =  0.0946,  and  hence  G  =  5°  24'. 

Since  the  angle  for  setting  a  milling-machine  table  is  calculated  to 
the  nearest  \°,  the  gashing  angle  for  the  worm-wheel  teeth  is  5  \°. 


WORM  GEARING 


115 


Cutting  Worm  Wheels.  When  cutting  worm  wheels  on  a 
milling  machine,  the  teeth  are  usually  gashed  first  and  then 
hobbed  to  give  them  their  final 
shape.  The  gashing  is  done 
with  an  involute  gear  cutter  of 
the  proper  number  and  pitch, 
the  table  of  the  machine  being 
first  adjusted  to  center  the 
blank  and  cutter  and  then  set 
to  the  gashing  angle. 

After  the  teeth  are  gashed 
the  milling-machine  table  is  set 
at  zero,  and  the  cutter  is  re- 
placed by  a  hob.  A  hob  is  a  tool 
shaped  like  a  worm  and  grooved 
to  form  cutting  teeth,  the  depth  GASHING  A  WORM  WHEEL  ON 

,.        i   .    T      .  T  T.^T  A  MILLING  MACHINE 

of  which  is  made  a  little  greater 

than  the  depth  of  the  worm  thread  to  allow  for  clearance. 


Exercises.   Worm  Gearing 

From  the  data  given,  make  the  calculations  for  cutting  each  of 
the  following  sets  of  worm  gearing : 

1.  Worm  wheel,  30  teeth;  worm,  single-threaded  with  a 
linear  pitch  of  0.3927". 

2.  Velocity  ratio,  20  to  1 ;  worm,  double-threaded  with  a 
linear  pitch  of  0.5236". 

Each  revolution  of  this  worm  will  move  the  worm  wheel  two  teeth. 

3.  Worm  wheel,  24  teeth;   worm,  single-threaded  with  a 
linear  pitch  of  0.2244". 

4.  Worm  wheel,  96  teeth;   worm,  triple-threaded  with  a 
lead  of  2.3562". 


116  GEAKS 

Exercises.    Gears 

1.  If,  in  a  spur  gear,  D0  =  2.228"  and  Pc  =  ^",  what  is 
the  number  of  teeth? 

2.  If  an  8-Pd  spur  gear  has  203  teeth,  what  is  D0  ? 

3.  Find  the  whole  depth  of  tooth  for  a  |-JJ  spur  gear. 

4.  Find  the  circular  pitch  for  a  32-T  spur  gear  in  which 
it  is  required  that  Dp— 16.55". 

5.  Two  8-JJ  bevel  gears  are  to  have  80  teeth  and  20  teeth 
respectively.    Make  the  calculations  for  cutting  the  gears. 

6.  If  a  20-inch  spur  gear  has  198  teeth,  what  is  the  pitch  ? 

7.  If  the  circular  pitch  of  a  spur  gear  is  2|",  what  is  the 
addendum?  the  whole  depth  of  tooth? 

8.  If  it  is  required  that  a  worm  wheel  is  to  have  20  teeth 
and  the  worm  a  single  thread  with  a  linear  pitch  of  0.3142", 
make  the  calculations  for  cutting  the  gearing. 

9.  If  a  39-T  spur  gear  has  a  pitch  of  31,  what  is  D0  ? 

10.  Make  the  necessary  calculations  for  cutting  a  pair  of 
spiral  gears,  it  being  required  that  Dc  (approximate)  =  12", 
Pn=6,  ^=30°,  ^=30°,  £=60°,  andr=3. 

11.  A  lf-^J  spur  gear  has   a  pitch  diameter  of  97.14". 
Find  the  outside  diameter  and  the  number  of  teeth. 

12.  Make  the  calculations  for  cutting  a  pair  of  bevel  gears, 
it  being  required  that  the  pitch  be  5,  the  number  of  teeth  on 
the  pinion  17,  and  the  velocity  ratio  3  to  1. 

13.  A  worm  wheel  is  to  have  60  teeth  and  is  to  mesh  with  a 
double-thread  worm  having  a  linear  pitch  of  1.0472".    Make 
all  the  calculations  for  cutting  the  gearing. 

14.  Find  the  pitch  diameters  of  two  spur  gears  in  mesh  if 
the  center  distance  between  the  gears  is  12",  and  one  gear 
has  40  teeth  and  the  other  20  teeth. 


CHAPTER  VII 

REVIEW  PROBLEMS 
Exercises.    General  Applications 

Change  the  following  tapers  per  inch  to  tapers  per  foot : 
1.  0.0762".       2.  0.08175".       3.  0.0162".       4.  0.0542". 

5.  If  the  outside  diameter  of  a  spur  gear  is  4J"  and  the 
pitch  is  18,  what  is  the  number  of  teeth  ? 

6.  Find  the  tap-drill  size  of  a  l|-inch  U.  S.  S.  thread. 

7.  If  a  quadruple-thread  screw  has  16  threads  per  inch, 
what  is  the  pitch  and  what  is  the  lead  ? 

8.  Find  the  change  gears  required  on  a  lathe,  which  has 
a  lead  screw  with  5  threads  to  the  inch,  when  cutting  a  metric 
thread  with  a  lead  of  1.5  mm. 

Assume  that  the  lathe  is  equipped  with  a  127-T  gear. 

9.  Two  12-7J  spur  gears  in  mesh  have  66  teeth  and  40 
teeth  respectively.    Find  the  center  distance. 

10.  Find  the  pitch  diameter  of  a  spur  gear  having  37  teeth 
and  a  pitch  of  10. 

11.  If  a  spiral  lead  of  27"  is  to  be  cut  on  a  J-inch  drill, 
to  what  angle  should  the  milling-machine  table  be  set  ? 

12.  Find  the  indexing  required  on  a  B.  &  S.  dividing  head 
for  milling  a  ratchet  wheel  with  grooves  13J°  apart. 

13.  Find  the  outside  diameter  of  a  96-T  spur  gear  which 
has  a  pitch  of  14. 

117 


118 


REVIEW  PROBLEMS 


Exercises.    Drill  Press 

1.  The  part  of  the  cast-iron  column  marked  A  in  the  figure 
below  is  2'  4"  long  and  was  turned  in  a  lathe  to  its  present 
diameter  of   6".     Find  how   long  it  took    to   machine  this 
casting  if  a  roughing  cut  with  a  |-inch  feed  and  a  finishing 
cut  with  a  ^-inch   feed  were 

taken  with  a  high-speed  tool. 

2.  The  finished   base  B  is 
12"  x  16"  and  was  planed  on 
a  3-to-l  planer  at  25  F.P.M., 
a  roughing  cut  with  a  y^-inch 
feed  and  a  finishing  cut  with  a 
|-inch  feed  being  taken.    Find 
the  length  of  time  consumed 
in  machining  this  casting. 

3.  The  diameter  of  the  driv- 
ing pulley   C  is  10  J"  at  the 
center  of  the  crown   and  10" 
at  the  edge,  and  the  width  of 
the  face  is  3".  Find  the  T.  P.  F. 
of  the  pulley. 

4.  What  was  the  tailstock 
offset  if  the  pulley  in  Ex.  3  was 
turned  on  a  7J-inch  arbor? 

5.  The  diameters  of  the  steps  of  the  cone  pulleys  are  5", 
7",  9",  and  11"  respectively,  and  pulley  C  makes  425  R.P.M. 
Beginning  with  the  belt  on  the  largest  step  of  the  driving 
pulley,  find  each  of  the  four  speeds  of  shaft  S. 

6.  Spindle  D  is  made  of  machine  steel  and  has  a  diameter 
of  llf"  and  a  length  of  28".    Find  the  number  of  R.P.M. 
of  the  lathe  for  a  roughing  cut  with  a  carbon-steel  tool. 


DRILL   PRESS 


DKILL  PEESS  119 

7.  The  spindle  in   Ex.  6  was  turned  from  round  stock, 
a  roughing  cut  with  a  Tlg-inch  feed  and  a  finishing  cut  with 
a  0.01 5-inch  feed  being  taken  with  a  high-speed  steel  tool. 
Find  the  time  taken  in  machining  the  spindle. 

8.  If  the   adjusting  screw  E  has  a  U.  S.  S.   thread   of 
Jg-inch  pitch,  what  is  the  root  diameter  of  the  screw  ? 

9.  Find  the  change  gears  required  on  a  lathe,  which  has 
a  lead  screw  with  5  threads  per  inch,  in  cutting  screw  E. 

10.  Bevel  gear  G  is  a  14-JJ  gear  with  51  teeth,  and  the 
pinion  meshing  with  it  has  17  teeth.    Make  all  the  necessary 
calculations  for  cutting  this  pair  of  gears. 

11.  If  spur  gear  E  has  71  teeth  and  a  circular  pitch  of 
0.3491",  what  is  its  outside  diameter? 

12.  In  Ex.  11  find  the   addendum  and  the  whole   depth 
of  tooth  of  gear  R. 

13.  Find  the  indexing  required  on  a  B.  &  S.  dividing  head 
in  cutting  gear  E. 

14.  Worm  W  is  a  single-thread  screw  with  a  linear  pitch 
of  0.2244",  and  meshes  with  a  worm  wheel  having  25  teeth. 
Make  the  calculations  for  cutting  the  worm  and  wheel. 

15.  If  the  adjusting  screw  E  has  an  Acme  screw  thread 
with  a  pitch  of  0.1111",  find  the  depth,  the  width  at  the 
top  of  the  thread,  and  the  number  of  threads  per  inch. 

16.  The  circular  table  of  the  drill  press  is  made  of  cast 
iron   and  has   a   diameter   of   27"  and  a  thickness  of   14". 

o 

Find  the  time  required  for  machining  the  circumference,  a 
roughing  cut  with  a  feed  of  0.05"  and  a  finishing  cut  with 
a  feed  of  0.125"  being  taken  with  a  high-speed  steel  tool. 

17.  Find  the  number  of  R.P.M.  for  boring  the  center  hole 
in  the  circular  table  in  Ex.  16  to  a  diameter  of  1^"  with  a 
carbon-steel  tool. 


120  EEVIEW  PROBLEMS 

Exercises.    Indexing  and  Milling 

1.  Using   a  Cincinnati   dividing  head,  find   the   indexing 
required  in  cutting  the  teeth  on  the  milling  cutter  shown 
in  the  first  figure  below. 

2.  In   cutting  the  teeth  on  -the   milling  cutter 
mentioned  in  Ex.  1  a  high-speed  steel  cutter  1J" 
in  diameter  was  used.  At  how  many  R.  P.  M.  should 
this  second  cutter  run  in  order  to  give  a  cutting 
speed  of  55  F.P.M.? 

3.  If  the  spiral  lead  of  the  teeth  is  to  be  26.25"  and  the 
diameter  of   the   blank   is   2J",   at  what  angle    should   the 
table  of  the  milling  machine  be  set  for  cutting 

the  spiral  grooves  in  the  milling  cutter  shown 
in  this  figure  ? 

4.  If  the  lead  of  the  milling  machine  in  Ex.  3 
is  10'-,  what  change  gears  should  be  used  ? 

5.  If  a  milling  cutter  similar  to  the  one  in  Ex.  3  is  to  be 
made  with  24  teeth,  find  the  indexing  required  on  a  B.  &  S. 
dividing  head  when  cutting  the  grooves. 

6.  Using  a  milling  machine  equipped  with   a  Cincinnati 
dividing  head,  find  the  indexing  required  in  milling  the  teeth 
in  the  metal-slitting  saw  here  shown. 

7.  If  a  metal-slitting   saw   similar   to    the 
one  mentioned  in  Ex.  6  is  to  be  made  with 
79   teeth,   find    the    indexing   required  on    a 
B.  &.  S.  dividing  head. 

8.  If  the  table  of  a  milling  machine  is  set  at  an  angle  of 
7|°  for  cutting  a  spiral  with  a  lead  of  62.50",  what  is  the 
diameter  of  the  work  ? 

9.  If  the  lead  of  the  milling  machine  in   Ex.  8   is   10", 
what  change  gears  should  be  used  ? 


INDEXING  AND  MILLING  121 

10.  Find   the  different  numbers   from  1  to  100  that  can 
be  indexed  by  simple  indexing  with  the  21-hole  circle  of  a 
B.  &  S.  dividing  head. 

11.  Using  a  Cincinnati  dividing  head,  find  the  indexing 
required  in  cutting  the  teeth  on  the  side-milling 

cutter  shown  in  this  figure. 

12.  If  the  side-milling  cutter  in  Ex.  11  is  4|" 
in  diameter,  what  is  the  cutting  speed  when  the 
cutter  is  driven  at  21  R.P.M.?  at  38  R.P.M.? 

Give  such  results  to  the  nearest  unit. 

13.  Using  a  Cincinnati  dividing  head,  find  the  indexing 
required   in    milling    the    screw-slotting    saw 

shown  in  this  figure,  the  saw  having  72  teeth. 

14.  Consider  Ex.  13  if  the  saw  is  to  have 
89  teeth  and  the  milling  machine  is  equipped 
with  a  B.  &  S.  dividing  head. 

15.  Using  compound  indexing  on  a  B.  &  S.  dividing  head, 
find  the  indexing  fractions  required  for  cutting  a  screw-slotting 
saw  which  is  to  have  93  teeth. 

16.  Find  the  change  gears  required  on  a  B.  &  S.  dividing 
head  for  indexing  ^°. 

It  is  necessary  to  index  for  4  x  360,  or  1440  divisions.  In  figuring 
the  ratio  of  the  change  gears  assume  a  movement  of  the  index  crank  of 
one  hole  on  the  33-hole  circle  for  each  division. 

17.  Find  all  the  circles  on  a  B.  &  S.  dividing  head  that 
will  index  in  common  the  numbers  3,  12,  15,  and  60. 

18.  What  is  the  lead  of  the  spiral  that  will  be  cut  on  a 
milling  machine  which  has  a  lead  of   10"  when  there  is  a 
56-T  gear  on  the  worm,  a  32-T  gear  as  the  first  gear  on  the 
stud,  a  48-T  gear  as  the  second  gear  on  the  stud,  and  a 
72-T  gear  on  the  feed  screw  ? 


122 


EEVIEW  PROBLEMS 


Exercises.    Tapers 

1.  Find  the  T.P.F.  of  taper  A  on  the  automobile  pinion 
shaft  shown  in  the  blueprint  on  page  123. 

2.  In  Ex.  1  find  the  tailstock  offset  for  cutting  taper  A. 

3.  As  in  Ex.  1,  find  the  T.P.F.  of  taper  B. 

4.  In  Ex.  3  find  the  tailstock  offset  for  cutting  taper  B. 

5.  Find  the  T.P.I,  of  the  gib  key  shown  in  the  blueprint. 

In  the  following  table  supply  each  missing  specification  for 
2-ball  clamping  levers  of  the  type  shown  in  the  blueprint : 


L 

i 

D 

d 

T.P.F. 

TAILSTOCK  OFFSET 

6. 

41" 

2ft" 

1" 

ft" 

7. 

5J 

8ft 

H 

i 

8. 

<* 

m 

I 

T9<f 

9. 

7* 

*H 

if 

5 
f 

10. 

8i 

5| 

it 

H 

11.  Find  the  T.P.F.  of  the  automobile  connecting  rod. 

In  the  following  table  supply  each  missing  specification  for 
tap-wrench  shells  of  the  type  shown  in  the  blueprint: 


D 

d 

? 

T.P.F. 

ANGLE  WITH  Axis 

12. 

ft" 

ft" 

M" 

13. 

1 

i   • 

1.6" 

14. 

J 

T6 

2.5 

15.  In  the  tapper  tap  blank  shown  in  the  blueprint,  if 
i  =  4£",  f  =  TV;,  £>  =  fV',  and  d  =  0.140",  find  the  T.P.F. 
and  the  included  angle  of  the  taper. 


TAPERS 


123 


124 


REVIEW  PROBLEMS 
Exercises.    Planer 


1.  The  table  A  of  the  planer  shown  in  the  figure  below  is 
4'  wide  and  18'  long  and  was  planed  on  a  2-to-l  planer  which 
had  a  cutting  speed  of  25  F.P.M.  If  a  roughing  cut  with  a 
^32 -inch  feed  and  a  finishing  cut  with  a  T7g-mch  feed  were 
taken,  how  long  did  it  take  to  machine  the  top  of  the  table  ? 


PLANER 


2.  The  V-shaped  ways  B  are  planed  to  an  angle  of  50°,  and 
the  width  of  the  V  at  the  top  is  2yy.   Find  the  depth  of  the  V. 

Use  the  formula  given  on  page  28,  the  included  angle  being  50°. 

3.  The  spur  gear   C  has   a  pitch  diameter  of  4"  and  a 
pitch  of  12.    Find  the  outside  diameter  of  the  gear. 

4.  In  Ex.  3  find  the  number  of  teeth  on  the  gear. 

5.  In  Ex.  3  find  the  whole  depth  of  a  tooth. 


PLANER  125 

6.  In  Ex.  3  find  the  indexing  required  in  cutting  the  gear 
on  a  milling  machine  equipped  with  a  B.  &  S.  dividing  head. 

7.  The  cast-iron  pulley  D  is  22"  in  diameter,  and  the  face 
is  3"  wide.    Find  the  number  of  R.P.M.  of  a  lathe  for  finish- 
ing the  face  with  a  high-speed  steel  tool. 

8.  In  Ex.  7  how  long  will  it  take  to  make  a  single  cut 
over  the  face  with  a  ^ -inch  feed  ? 

9.  Find  the  depth  of  the  thread  on  the  feed   screw  E 
which  has  6  U.  S.  S.  threads  to  the  inch. 

10.  In  Ex.  9  find  the  change  gears  required  on  a  lathe  with 
a  lead  screw  of  ^-inch  pitch  when  cutting  the  thread. 

11.  The  14-^  spur  gear  F  has  57  teeth.    Find  the  outside 
diameter  of  the  gear. 

12.  In  Ex.  11  find  the  thickness  of  tooth  on  the  pitch 
line  and  also  find  the  working  depth  of  a  tooth. 

13.  In  Ex.  11  find  the  indexing  required  on  a  B.  &  S. 
dividing  head  in  cutting  the  gear. 

Why  will  two  intermediate  gears  be  needed  ? 

14.  If  the  crowned  pulley  G  has  a  6i-inch  face  and  is  T5g" 
larger  in  diameter  at  the  center  of  the  crown  than  at  the  edges, 
find  the  T.P.F.  of  the  pulley. 

15.  If  pulley  G  is  turned  on  a  Ill-inch  mandrel,  find  the 
tailstock  offset. 

16.  If  the  cast-iron  pulley  H  has  a  diameter  of  28 |-"  and 
a  3-inch  face,  find  the  number  of  R.P.M.  for  roughing  the 
face  with  a  tool  of  carbon  steel. 

17.  In  Ex.  16  how  long  will  it  take  to  make  a  single  cut 
over  the  face  with  a  ^-inch  feed  ? 

18.  Find  the  number  of  R.P.M.  of  pulley  H  if  it  is  belted 
to  a  24-inch  pulley  which  has  a  speed  of  125  R.P.M. 


126 


REVIEW  PROBLEMS 
Exercises.    Drilling  Machine 


1.  If  motor  A  in  the  figure  below  makes  750  R.P.M.,  if 
gear  1  on  the  motor  has  24  teeth,  and  if  gear  2  has  56  teeth, 
what  is  the  speed  of  the  shaft  to  which  gear  2  is  fastened? 


WALL   RADIAL-DRILLING    MACHINE 


2.  If  the  gears  in  Ex.  1  were  spur  gears  with  a  pitch  of  14, 
what  would  be  the  respective  outside  diameters  ? 

3.  What  would    be  the   center  distance  if  gears  1  and  2 
were  spur  gears  which  meshed  together  instead  of  being  con- 
nected by  the  endless  chain  ? 

4.  Find  the    indexing  required  on  a  Cincinnati  dividing 
head  for  cutting  the  gears  in  Ex.  3. 

5.  The  6-7J  bevel   gears  B  have   48   teeth  and  16  teeth 
respectively.    Make  the  calculations  for  cutting  these  gears. 


DRILLING  MACHINE  127 

6.  The  socket  in  spindle  C  is  bored  to  a  #4  Morse  taper, 
which  has  a   T.P.F.    of   0.623".    To   what   angle   should   a 
compound  rest  be  set  to  bore  this  taper? 

7.  Shaft  Z>,  which  is  made  of  machine  steel,  is  10'  6"  long 
and  has  a  diameter  of  If".    Find  the  number  of  R.P.M.  of 
a*  lathe  for  roughing  the  shaft  with  a  high-speed  steel  tool. 

8.  In  Ex.  7  find  how  long  it  will  take  to  make  a  single 
cut  over  the  shaft  with  a  feed  of  0.015". 

9.  The  worm  wheel  G  has  60  teeth  and  it  meshes  with 
a  single-thread  worm,  the  linear  pitch  of  which  is  0.1963". 
Make  the  calculations  for  cutting  the  worm  and  wheel. 

10.  If  the  weight  H  is  made  of  cast  iron  and  is  approxi- 
mately 2|-"  x  5|"  x  8^-",  find  its  approximate  weight. 

The  weight  of  cast  iron  may  be  taken  as  450  Ib.  per  cubic  foot. 

11.  If  the  diameter  at  the  middle  of  the  l|-inch  face  of 
the  cast-iron  wheel  J  is  4||"  and  that  at  the  edge  is  4|",  what 
is  the  T.P.F.  of  the  wheel? 

12.  Using  the  smaller  diameter  in  Ex.  11,  at  what  speed 
should  the  face  be  finished  with  a  high-speed  tool  ? 

13.  In  Ex.  11  find  the  time  required  for  finishing  25  wheels 
of  this  type,  making  a  single  cut  with  a  gL-mch  feed  over  the 
face  of  each  wheel. 

14.  Shaft  D  makes  an  angle  of  22°  with  brace  E  and  is 
perpendicular  to  shaft  F.    If  the  distance  between  the  axis 
of  shaft  F  and  the  vertex  of  the  angle  formed  by  Z>  and  E 
is  9'  8",  find  to  the  nearest  0.1"  the  length  of   E  between 
the  vertices  of  the  angles  formed  by  E  with  D  and  F. 

The  ratio  D  :  E  is  what  function  of  the  angle  between  them  ?  This 
example  and  Exs.  3-22  on  page  134  may  be  omitted  by  those  students 
who  have  not  had  the  equivalent  of  Chapter  IV  (Trigonometry)  in 
"  Fundamentals  of  Practical  Mathematics  "  in  this  series. 


128 


REVIEW  PROBLEMS 


Exercises.    Universal  Milling  Machine 

1.  In   the  universal  milling  machine   shown   below  the 
diameter  of  the  machine-steel  arm  A  is  4J"  and  the  length 
is  3'  2".    Find  the  number 

of  R.P.M.  for  a  roughing 
cut  with  a  high-speed  tool. 

2.  Taking   a  roughing 
cut  with  a  ^-inch  feed  and 
a  finishing  cut  with  a  -j5^ 
inch  feed,  how  long  will  it 
take  to  machine  the   arm 
in  Ex.  1  ? 


3.  If  table  B  is 
wide  and  39"  long,  how 
long  will  it  take,  using  a 
Jg-inch  feed,  to  plane  the 
top  of  the  table  on  a  2-to-l 
planer  which  has  a  cutting 
speed  of  25  F.P.M.? 


UNIVERSAL   MILLING   MACHINE 


If  spur  gear  C  has  an  outside  diameter  of  8j"  and  a  pitch  of 
12,  find  each  -of  the  following  : 

4.  Number  of  teeth.  6.  Addendum  and  clearance. 

51.  Circular  pitch.  7.  Whole  depth  of  tooth. 

8.  In  the  taper  hole*  in  the  spindle  D  the  large  diameter 
is  1.260",  the  small  diameter  is  1.045",  and  the  length  of 
the  taper  is  5".    Find  the  T.P.F. 

9.  In  Ex.  8  find  the  included  angle  of  the  taper. 

10.  Screw  E,  which  regulates  the  vertical  adjustment  of 
the  table,  has  a  double  Acme  screw  thread  of  ^-inch  lead. 
Find  the  double  depth  of  the  thread. 


UNIVERSAL  MILLING  MACHINE  129 

11.  In  Ex.  10  find  the  change  gears  required  on  a  lathe 
with  a  lead  screw  of  1-inch  pitch  when  cutting  the  thread. 

In  cutting  a  double  thread  the  gears  are  figured  as  if  for  a  single 
thread  of  the  required  lead.  One  groove  is  cut  to  the  double-thread 
depth,  the  work  is  given  a  half  turn,  and  then  the  other  groove  is  cut. 

12.  If  gear  F  has  44  teeth,  find  the  indexing  required  on 
a  B.  &  S.  dividing  head  in  cutting  the  gear. 

13.  If  the  diameters  of  the  steps  of  the  cone  pulley  H  are 
7£",  8-f",  101",  and  12"  respectively,  and  the  pulley  is  made 
of  cast  iron,  find  the  number  of  R.P.M.  needed  in  roughing 
the  face  of  each  step  with  a  tool  of  carbon  steel. 

14.  If  each  step  of  the  cone  pulley  in  Ex.  13  has  a  21-inch 
face,  find  the  time  taken  in  making  a  roughing  cut  over  the 
four  faces  with  a  -jL-inch  feed.. 

15.  Beginning  with  the  smallest  step,  a  finishing  cut  was 
taken    over  the  faces  of  the   steps    of   the  cone  pulley  in 
Ex.  13,  the  speeds  of  the  lathe  being  as  follows:  70  R.P.M., 
60  R.P.M.,  50  R.P.M.,  45  R.P.M.    Find  the  average  cut- 
ting speed  used  in  finishing  the  pulley. 

16.  If  the  diameter  at  the  center  of  each  step  of  the  cone 
pulley  in  Ex.  13  is  -fa11  greater  than  that  at  the  edge,  find 
the  T.P.F.  of  the  face. 

17.  The  cone  pulley  in  Ex.  13  is  driven  by  a  similar  cone 
pulley  on  a  countershaft  which  makes  98  R.P.M.,  the  7^-inch 
steps  being  opposite  the  12-inch  steps.    Beginning  with  the 
lowest  speed,  find  the  speed  of  the  shaft  to  which  cone  pulley 
H  is  fastened  for  each  position  of  the  belt  on  the  steps. 

18.  If  the  outside  diameter  of  the  nose  of  the  spindle  D  is 
2^-"  and  the  nose  is  threaded  with  a  sharp  V-thread  of  ^-inch 
pitch,  to  what  diameter  should  the  collar  G  be  bored? 

19.  If  the  lead  of  the  milling  machine  is  10",  what  change 
gears  are  needed  in  cutting  a  spiral  with  a  lead  of  2.24"? 


130 


REVIEW  PROBLEMS 


Exercises.    Tapers 

1.  Find  the  T.P.F.  of  the  taper  on  the  automobile  stub 
axle  shown  in  the  blueprint  on  page  131. 

2.  In  Ex.  1  find  the  tailstock  offset  for  turning  the  taper. 

In  the  following  table  supply  each  missing  specification  for 
automobile  stub  axles  of  the  type  shown  in  the  blueprint  : 


3. 

4. 
5. 


L 

I 

D 

d 

T.P.F. 

TAILSTOCK  OFFSET 

11J" 

3" 

if" 

1.500" 

9f 

IT"* 

H" 

1.250 

!<>A 

2T9e 

1* 

1.4 

In  the  following  table  supply  each  missing  specification  for 
ball-crank  handles  of  the  type  shown  in  the  blueprint  : 


6. 

7. 
8. 
9. 


L 

I 

D 

d 

T.  P.  F. 

TAILSTOCK  OFFSET 

3" 

If" 

|W 

* 

iV' 

31 

H 

& 

A 

3| 

Hi 

M 

44 

4 

8* 

H 

4f 

In  the  following  table  supply  each  missing  specification  for 
bench-lathe  drawback  chucks  of  the  type  shown  in  the  blueprint : 


10. 
11. 
12. 
13. 


/ 

D 

d 

ANGLE  WITH  Axis 

T.P.F. 

0.300" 

0.500" 

0.335" 

0.850 

0.650 

12° 

1.156 

0.950 

2.125" 

1.125 

1.875 

1.625 

TAPERS 


131 


132 


KEVIEW  PROBLEMS 
Exercises.    Engine  Lathe 


1.  In  the  lathe  here  shown  the  cast-iron  faceplate  A  has 
a  diameter  of  14",  and  the  width  of  the  face  is  1|".  Using  a 
carbon-steel  tool,  at  what  speed  should  the  face  be  finished  ? 


ENGINE    LATHE 


2.  In  Ex.  1  how  long  will  it  take,  using  a  gL-inch  feed,  to 
make  a  single  cut  over  the  face  ? 

3.  If  the  lead  screw  B  has  an  Acme  screw  thread  with 
8  threads  to  the  inch,  find  the  depth  of  the  thread. 

In  the  screw  described  in  Ex.  3  find  each  of  the  following  : 

4.  The  width  of  the  space  at  the  top  of  the  thread. 

5.  The  width  of  the  flat  at  the  root  of  the  thread. 

6.  The  width  of  the  flat  at  the  top  of  the  thread. 

7.  The  change  gears  required  on  a  lathe,  which  has  a  lead 
screw  with  5  threads  per  inch,  in  cutting  the  thread. 


ENGINE  LATHE  133 

8.  Spur  gear  C  is  a  12-J^  gear  with  96  teeth.    Find  the 
pitch  diameter  of  the  gear. 

9.  In  Ex.  8  find  the  outside  diameter  of  the  gear. 

10.  In  Ex.  8  find  the  whole  depth  of  tooth. 

11.  The  dead  center  D  is  1.475"  in  diameter  at  its  small 
end,  1.790"  in  diameter  at  its  large  end,  and  the  length  of 
the  taper  is  6".    Find  the  T.P.F. 

12.  In  Ex.  11  find  the  included  angle  of  the  taper. 

13.  In  the  cast-iron   cone   pulley  E  the  diameters  of   the 
steps  are  61",  7T\",  8|",  91f",  and  11"  respectively,  and  the 
width  of  each  face  is  1|".    Beginning  with  the  smallest  step, 
find  the  number  of  R.P.M.  for  finishing  each  face  with  a 
high-speed  steel  tool. 

14.  If  the  diameters   at  the   centers  of  the  steps  of  the 
pulley  in  Ex.  13  are  ^V  greater  than  those  at  the  edges,  find 
the  angle  which  the  faces  make  with  the  axis  of  the  pulley. 

15.  If  the  cone  pulley  in  Ex.  13  is  turned  on  a  7^-inch 
arbor,  find  the  offset  of  the  tailstock  center. 

16.  If  the  machine-steel  feed  rod  F  is  {§"  *n  diameter  and 
5' 8"  in  length,  what  is  the  number  of  R.P.M.  needed  in 
roughing  the  rod  with  a  tool  of  carbon  steel? 

17.  In  Ex.  16  how  long  will  it  take  to  make  a  single  cut 
over  the  rod  with  a  I -inch  feed  ? 

o 

18.  If  the  spur  gear  G  has  117  teeth  and  the  pitch  is  8, 
what  is  the  outside  diameter? 

19.  In  Ex.  18  find  the  circular  pitch  and  the  thickness  of 
tooth  on  the  pitch  line. 

20.  Using   a  B.  &  S.    dividing    head,    find    the    indexing 
required  in  cutting  the  gear  in  Ex.  18. 

21.  A   piece    of   cast   iron   1^"   in   diameter  is   turned   in 
the  lathe  at  175  R.P.M.    Find  the  cutting  speed  used. 


134  REVIEW  PROBLEMS 

Exercises.    General  Applications 

1.  Find   the   angle  a  in  the   lathe  center  shown   in  the 
blueprint  on  page  135. 

2.  In  Ex.  1  find  the  length  L  of  the  pointed  end. 

3.  The  three  holes  J,  B,  C  are  to  be  drilled  on  a  milling 
machine  in  the  jig  plate  shown  in  the  blueprint.    Find  to  the 
nearest    0.001"  the  vertical    movement  from  A  for  drilling 
hole  C.   Find  the  lateral  movement  from  C  for  drilling  hole  B. 

In  connection  with  Exs.  3-22  see  the  note  at  the  foot  of  page  127. 

4.  The  holes  in  the  cylinder  head  shown  in  the  blueprint 
are  equally  spaced  on  the  circle.    Find  the  straight-line  dis- 
tance between  the  centers  of  the  holes. 

5.  Consider  Ex.  4  if  there  are  9  holes  on  a  12-inch  circle ; 
13  holes  on  an  18-inch  circle ;  21  holes  on  a  16-inch  circle. 

6.  In  the   pulleys  and  belting   shown   in   the   blueprint 
find  the  angle  a  and  also  find  the  distance  D  between  the 
centers  of  the  pulleys. 

7.  Find  the  diameter   of   the   round   stock  required  for 
making  hexagonal  heads  which  are  to  be  |"  across  the  flats. 

Consider  Ex.  7  for  each  of  the  following  measurements  across 
the  flats : 

8.  If".         9.   lTy.       10.   1J".          11.   111".       12.   2T3/. 

Find  in  each  case  the  largest  square  that  can  be  milled  on 
a  round  shaft  in  which  the  diameter  is  as  follows : 

13.  Ty.       14.  ||".         15.  lTy.       16.   lif  ".       17.  25y. 

Find  in  each  case  the  diameter  of  the  round  stock  required  in 
milling  a  square  in  which  the  length  of  side  is  as  follows  : 

18.  f".          19.  11".         20.   lTy.       21.   If".         22.   2|". 


GEKEKAL  APPLICATIONS 


135 


136  REVIEW  PROBLEMS 

Exercises.    Miscellaneous  Problems 

1.  A  plug  gage  T9g"  in  diameter  enters  a  hole  with  un- 
known taper  a  distance  of  3^",  and  another  plug  gage  11" 
in  diameter  enters  the  same  hole  a  distance  of  2J".    Find 
the  T.P.F.  of  the  hole. 

2.  Find  the  change  gears  required  on  a  milling  machine 
with  a  lead  of  10"  in  cutting  a  spiral  with  a  lead  of  12.90". 

3.  Make  the  necessary  calculations  for  cutting  a  pair  of 
16-^  bevel  gears  if  the  velocity  ratio  of  the  gears  is  to  be 
3  to  4,  if  the  shafts  are  to  be  at  right  angles  to  each  other, 
and  if  the  pinion  is  to  have  48  teeth. 

4.  If  a  milling  machine  is  geared  to  cut  a  spiral  with  a 
lead  of   0.67",  find   the   angle   at  which  the  dividing  head 
should  be  elevated  in  milling  a  cam  which  is  to  have  a  rise 
of  0.310"  in  0.47  of  the  circumference. 

5.  The  center  of  a  12-7J  spur  gear  with  54  teeth  is  8|" 
from  the  center  of  a  mating  gear.  Find  the  number  of  teeth 
and  the  outside  diameter  of  the  second  gear. 

6.  Make  all  the  necessary  calculations  for  cutting  a  pair 
of  spiral  gears  if  it  is  required  that  Dc  (approximate)  =  12", 
r  =  2,  Ag  =  33°,  Ap  =  27°,  Pn  =  8,  and  S  =  60°. 

7.  Two  spur  gears  are  to  mesh  with  a  distance  of  27" 
between  centers,  and  the  velocity  ratio  is  to  be  1  to  2.    If 
the  gears  are  to  be  cut  10  pitch,  what  should  be  the  outside 
diameter  and  the  number  of  teeth  on  each  gear? 

8.  If  an  Acme  screw  thread  with  3  threads  per  inch  is  cut 
on  a  2^-inch  shaft,  what  is  the  root  diameter  of  the  thread  ? 

9.  The  tailstock  center  of  a  certain  lathe  can  be  offset  a 
maximum  distance  of  21"  from  the  center  line  of  the  lathe. 
What  is  the  greatest  T.P.F.  that  can  be  cut  on  a  piece  of 
work  9"  long?  on  a  piece  of  work  17"  long? 


TABLES  FOR  REFERENCE 

TABLES  OF  MEASURES 

LENGTH 
12  inches  (in.  or  ")  =  1  foot  (ft.  or  ') 

3  feet  =  1  yard  (yd.) 
5|  yards,  or  16|  feet  =  1  rod  (rd.) 
320  rods,  or  5280  feet  =  1  mile  (mi.) 

SQUARE  MEASURE 
144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.) 

9  square  feet  =  1  square  yard  (sq.  yd.) 
30|  square  yards  =  1  square  rod  (sq.  rd.) 
160  square  rods  =  1  acre  (A.) 

CUBIC  MEASURE 

1728  cubic  inches  (cu.  in*.)  =  1  cubic  foot  (cu.  ft.) 
27  cubic  feet  =  1  cubic  yard  (cu.  yd.) 
128  cubic  feet  =  1  cord  (cd.) 

WEIGHT 

16  ounces  (oz.)  =  1  pound  (Ib.) 
2000  pounds  =  1  ton  (T.) 

LIQUID  MEASURE 
4  gills  (gi.)  =  1  pint  (pt.) 

2  pints  =  1  quart  (qt.) 
4  quarts  =  1  gallon  (gal.) 

DRY  MEASURE 
2  pints  (pt.)  =  1  quart  (qt.) 
8  quarts  =  1  peck  (pk.) 
4  pecks  =  1  bushel  (bu.) 

A  bushel  contains  2150.42  cu.  in.  (approximately  l£  cu.  ft.). 

137 


138  TABLES  FOB  REFERENCE 

METRIC  LENGTH 

1  kilometer  (km.)  =  1000  meters 

Meter  (m.) 

1  centimeter  (cm.)  =0.01  meter 
1  millimeter  (mm.)  =0.001  meter 

In  comparing  the  metric  measures  of  length  with  those  of  our 
common  system  we  usually  think  of  1  km.  as  2  mi.,  of  1  m.  as  39^",  of 
1  cm.  as  f",  or  0.4",  and  of  1  mm.  as  ^V  >  or  0.04".  For  a  higher  degree 
of  accuracy,  however,  we  use  the  following  approximate  equivalents : 
1  km.  =  0.62  mi.,  1  m.  =  39.37",  1  cm.  =  0.3937",  and  1  mm.  =  0.0394". 
To  the  machinist  the  millimeter  is  the  most  important  of  the  metric 
measures  on  account  of  its  use  in  connection  with  imported  automobiles 
and  with  machines  made  for  foreign  trade. 

METRIC  WEIGHT 

1  metric  ton  (t.)  =  1000  kilograms 
1  kilogram  (kg.)  =  1000  grams 
Gram  (g.) 

The  following  approximate  equivalents  may  be  used  in  compar- 
ing metric  measures  of  weight  with  those  of  our  common  system  : 
1 1.  =  2204.6  lb.,  1  kg.  =  2.2  lb.,  and  1  g.  =  15.43  grains. 

METRIC  CAPACITY 

1  hektoliter  (hi.)  =  100  liters 

Liter  (1.) 
1  centiliter  (cl.)=  0.01  liter 

The  liter  is  approximately  a  quart.  It  is  the  volume  of  a  cube  that 
is  0.1  m.,  or  about  4",  on  an  edge. 

ANGLES  AND  ARCS 

60  seconds  (")  =  1  minute  (') 
60  minutes  =  1  degree  (°) 
90  degrees  =  1  right  angle 
360  degrees  =  1  circumference 


TABLES  FOE,  REFERENCE  139 

DECIMAL  EQUIVALENTS  OF  COMMON  FRACTIONS 


FRACTION 

DKCIMAL 

FRACTION 

DECIMAL 

A 

0.015625 

ii 

0.515625 

1 
38 

.03125 

ii 

.53125 

3_ 

.046875 

35 

.546875 

_i 

.0625 

T9(f 

.5625 

A 

.078125 

37 

.578125 

3 
32 

.09375 

Si 

.59375 

TT74 

.109375 

II 

.609375 

1 
8 

.125 

5 
8 

.625 

o94 

.140625 

ti 

.640625 

5 
T52 

.15625 

H 

.65625 

1  1 
04 

.171875 

43 
0  4 

.671875 

T3(T 

.1875 

11 

.6875 

H 

.203125 

H 

.703125 

V<>- 

.21875 

If 

.71875 

U 

.234375 

ti 

.734375 

1 

.25 

3 

.75 

4 

4 

iJ 

.265625 

H 

.765625 

_9 

.28125 

14 

.78125 

i  j> 

.296875 

H 

.796875 

tV 

.3125 

ii 

.8125 

04 

.328125 

H 

.828125 

U 

.34375 

II 

.84375 

H 

.359375 

5  5 
?J4 

.859375 

3 

8 

.375 

7 
8 

.875 

If 

.390625 

ft 

.890625 

i  a 

.40625 

^ 

.90625 

2  7 
04 

.421875 

H 

.921875 

7 

.4375 

1  5 

.9375 

it 

.453125 

1  0 

li 

.953125 

M 

.46875 

U 

.96875 

in 

.484375 

11 

.984375 

2 

.5 

1 

1. 

140  TABLES  FOR  REFERENCE 

CONVENIENT  RULES 

The  circumference  of  a  circle  is  ^  times  the  diameter. 
For  a  higher  degree  of  accuracy,  c  =  3.1416  d. 

The  diameter  of  a  circle  is  -j^  of  the  circumference. 
For  a  higher  degree  of  accuracy,  d  =  0.3183  c. 

The  area  of  a  circle  is  j^  of  the  square  of  the  diameter. 
For  a  higher  degree  of  accuracy,  A  =  0.7854  cP. 

The  height  of  an  equilateral  triangle  is  0.8660  of  the  side. 
The  diagonal  of  a  square  is  1.4142  times  the  side. 

The  diagonal  of  a  square  is  also  called  the  "  long  diameter  "  or  the 
"  distance  across  the  corners." 

The  long  diameter  of  a  regular  hexagon  is  twice  the  side. 

The  short  diameter,  or  the  perpendicular  distance  between 
parallel  sides,  of  a  regular  hexagon  is  1.7321  times  the  side. 

To  convert  Fahrenheit  temperature  into  centigrade  temper- 
ature subtract  32°  from  the  Fahrenheit  reading  and  take  ^  of 
the  result. 

Expressed  as  a  formula,  C  =  f  (F  —  32). 

To  convert  centigrade  temperature  into  Fahrenheit  temperature 
take  j?  of  the  centigrade  reading  and  add  32°  to  the  result. 
Expressed  as  a  formula,  F  =  f  C  +  32. 

COMMON  EQUIVALENTS 

In  ordinary  cases  the  following  equivalents  may  be  used: 

1  gal.  contains  231  cu.  in.,  or  0.134  cu.  ft. 

1  cu.  ft.  contains  7|  gal. 

1  barrel  (bbl.)  contains  3l|gal.,  or  4|cu.  ft. 

1  cu.  ft.  of  water  weighs  62.425  Ib.  (approximately  62 1  lb.). 

1  gal.  of  water  weighs  8.345  lb.  (approximately  s|  lb.). 

1 1.  of  water  weighs  1  kg. 


TABLES  FOE  BEFEKENCE  141 

NATURAL  TRIGONOMETRIC  FUNCTIONS 

Tables  of  four  of  the  natural  functions  of  an  angle,  the 
sine,  cosine,  tangent,  and  cotangent,  for  every  6',  or  0.1°,  from 
0°  to  90°  are  given  on  pages  142-149. 

The  following  examples  illustrate  the  use  of  these  tables : 

1.  Find  sin  62°  19'. 

On  page  143  we  look  for  62°  in  the  column  at  the  left,  and  in  the 
same  line  at  the  right  under  18'  we  find  that 

sin  62°  18' =  0.8854. 

Under  V  in  the  column  of  differences  at  the  right  and  in  line  with 
62°  we  find  that  the  difference  for  1'  is  1  (actually  this  is  0.0001). 

Since  the  column  of  differences  is  marked  "+  Differences,"  this  differ- 
ence is  added  to  sin  62°  18'.  The  addition  is  done  mentally,  only  the 
result  being  written.  Therefore 

sin  62°  19' =  0.8855. 

2.  Find  cot  5°  56'. 

From  page  148,  cot  5°  54'  =  9.6768. 

Notice  that  after  the  5  at  the  left  the  integral  part  is  11,  but  since 
two  black  numbers  intervene,  we  decrease  the  11  by  2.  The  cotangent 
is  here  changing  so  rapidly  that  we  use  a  process  commonly  known 
as  interpolation.  By  subtraction  we  find  that  the  difference  between 
cot  5°  54'  and  cot  6°  0'  is  0.1624.  Since  2'  is  ^  of  6',  we  take  £  of  0.1624, 
and  hence  the  difference  for  2'  is  0.0541. 

Since  the  column  of  differences  is  marked  te—  Differences,"  this  differ- 
ence is  subtracted  from  cot  5°  54'.  Therefore 

cot  5°  56'=  9.6227. 

3.  Find  the  angle  of  which  the  tangent  is  0.5693. 

On  page  146  we  find  that  the  tangent  next  smaller  than  this  tangent 
is  0.5681,  which  is  the  tangent  of  29°  36',  and  by  subtraction  we  see 
that  the  difference  between  0.5693  and  0.5681  is  0.0012. 

In  the  column  of  differences  we  see  that  12  is  the  difference  to  be 
added  for  3'.  Adding  3'  to  29°  36',  we  see  that 

29°  39'  is  the  angle  of  which  the  tangent  is  0.5693. 


142 


NATURAL  SINES.    0°-45° 


0 

0.0° 

0.1° 

0.2° 

0.3° 

0.4° 

0.5° 

0.6° 

0.7° 

0.8° 

0.9° 

+  Differences 

0' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48' 

54' 

V 

2' 

3' 

4' 

5' 

~o 

.0000 

.0017 

.0035 

.0052 

.0070 

.0087 

.0105 

.0122 

.0140 

.0157 

3 

6 

9 

12 

15 

1 

.0175 

.0192 

.0209 

.0227 

.0244 

.0262 

.0279 

.0297 

.0314 

.0332 

3 

6 

9 

12 

15 

2 

.0349 

.0366 

.0384 

.0401 

.0419 

.0436 

.0454 

.0471 

.0488 

.0506 

3 

6 

9 

12 

15 

3 

.0523 

.0541 

.0558 

.0576 

.0593 

.0610 

.0628 

.0645 

.0663 

.0680 

3 

6 

9 

12 

15 

4 

.0698 

.0715 

.0732 

.0750 

.0767 

.0785 

.0802 

.0819 

.0837 

.0854 

3 

6 

9 

12 

14 

5 

.0872 

.0889 

.0906 

.0924 

.0941 

.0958 

.0976 

.0993 

.1011 

.1028 

3 

6 

9 

12 

14 

6 

.1045 

.1063 

.1080 

.1097 

.1115 

.1132 

.1149 

.1167 

.1184 

.1201 

3 

6 

9 

12 

14 

7 

.1219 

.1236 

.1253 

.1271 

.1288 

.1305 

.1323 

.1340 

.1357 

.1374 

3 

6 

9 

12 

14 

8 

.1392 

.1409 

.1426 

.1444 

.1461 

.1478 

.1495 

.1513 

.1530 

.1547 

3 

6 

9 

12 

14 

9 

.1564 

.1582 

.1599 

.1616 

.1633 

.1650 

.1668 

.1685 

.1702 

.1719 

3 

6 

9 

12 

14 

10 

.1736 

.1754 

.1771 

.1788 

.1805 

.1822 

.1840 

.1857 

.1874 

.1891 

3 

6 

9 

11 

14 

11 

.1908 

.1925 

.1942 

.1959 

.1977 

.1994 

.2011 

.2028 

.2045 

.2062 

3 

6 

9 

11 

14 

12 

.2079 

.2096 

.2113 

.2130 

.2147 

.2164 

.2181 

.2198 

.2215 

.2233 

3 

6 

9 

11 

14 

13 

.2250 

.2267 

.2284 

.2300 

.2317 

.2334 

.2351 

.2368 

.2385 

.2402 

3 

6 

8 

11 

14 

14 

.2419 

.2436 

.2453 

.2470 

.2487 

.2504 

.2521 

.2538 

.2554 

.2571 

3 

6 

8 

11 

14 

15 

.2588 

.2605 

.2622 

.2639 

.2656 

.2672 

.2689 

.2706 

.2723 

.2740 

3 

6 

8 

11 

14 

16 

.2756 

.2773 

.2790 

.2807 

.2823 

.2840 

.2857 

.2874 

.2890 

.2907 

3 

6 

8 

11 

14 

17 

.2924 

.2940 

.2957 

.2974 

.2990 

.3007 

.3024 

.3040 

.3057 

.3074 

3 

6 

8 

11 

14 

18 

.3090 

.3107 

.3123 

.3140 

.3156 

.3173 

.3190 

.3206 

.3223 

.3239 

3 

6 

8 

11 

14 

19 

.3256 

.3272 

.3289 

.3305 

.3322 

.3338 

.3355 

.3371 

.3387 

.3404 

3 

5 

8 

11 

14 

20 

.3420 

.3437 

.3453 

.3469 

.3486 

.3502 

.3518 

.3535 

.3551 

.3567 

3 

5 

8 

11 

14 

21 

.3584 

.3600 

.3616 

.3633 

.3649 

.3665 

.3681 

.3697 

.3714 

.3730 

3 

5 

8 

11 

14 

22 

.3746 

.3762 

.3778 

.3795 

.3811 

.3827 

.3843 

.3859 

.3875 

.3891 

3 

5 

8 

11 

14 

23 

.3907 

.3923 

.3939 

.3955 

.3971 

.3987 

.4003 

.4019 

.4035 

.4051 

3 

5 

8 

11 

14 

24 

.4067 

.4083 

.4099 

.4115 

.4131 

.4147 

.4163 

.4179 

.4195 

.4210 

3 

5 

8 

11 

13 

25 

.4226 

.4242 

.4258 

.4274 

.4289 

.4305 

.4321 

.4337 

.4352 

.4368 

3 

5 

8 

11 

13 

26 

.4384 

.4399 

.4415 

.4431 

.4446 

.4462 

.4478 

.4493 

.4509 

.4524 

3 

5 

8 

10 

13 

27 

.4540 

.4555 

.4571 

.4586 

.4602 

.4617 

.4633 

.4648 

.4664 

.4679 

3 

5 

8 

10 

13 

28 

.4695 

.4710 

.4726 

.4741 

.4756 

.4772 

.4787 

.4802 

.4818 

.4833 

3 

5 

8 

10 

13 

29 

.4848 

.4863 

.4879 

.4894 

.4909 

.4924 

.4939 

.4955 

.4970 

.4985 

3 

5 

8 

10 

13 

30 

.5000 

.5015 

.5030 

.5045 

.5060 

.5075 

.5090 

.5105 

.5120 

.5135 

3 

5 

8 

10 

13 

31 

.5150 

.5165 

.5180 

.5195 

.5210 

.5225 

.5240 

.5255 

.5270 

.5284 

2 

5 

7 

10 

12 

32 

.5299 

.5314 

.5329 

.5344 

.5358 

.5373 

.5388 

.5402 

.5417 

.5432 

2 

5 

7 

10 

12 

33 

.5446 

.5461 

.5476 

.5490 

.5505 

.5519 

.5534 

.5548 

.5563 

.5577 

2 

5 

7 

10 

12 

34 

.5592 

.5606 

.5621 

.5635 

.5650 

.5664 

.5678 

.5693 

.5707 

.5721 

2 

5 

7 

10 

12 

35 

.5736 

.5750 

.5764 

.5779 

.5793 

.5807 

.5821 

.5835 

.5850 

.5864 

2 

5 

7 

9 

U2 

36 

.5878 

.5892 

.5906 

.5920 

.5934 

5948 

5962 

.5976 

.5990 

.6004 

2 

5 

7 

9 

12 

37 

.6018 

.6032 

.6046 

.6060 

.6074 

6088 

6101 

.6115 

.6129 

.6143 

2 

5 

7 

9 

12 

38 

.6157 

.6170 

.6184 

.6198 

.6211 

6225 

6239 

.6252 

.6266 

.6280 

2 

5 

7 

9 

11 

39 

.6293 

.6307 

.6320 

.6334 

.6347 

6361 

6374 

.6388 

.6401 

.6414 

2 

4 

7 

9 

11 

40 

.6428 

.6441 

.6455 

.6468 

.6481 

6494 

6508 

.6521 

.6534 

6547 

2 

4 

7 

9 

11 

41 

.6561 

.6574 

.6587 

.6600 

.6613 

6626 

6639 

.6652 

6665 

.6678 

2 

4 

7 

9 

11 

42 

.6691 

.6704 

.6717 

.6730 

.6743 

6756 

6769 

.6782 

.6794 

.6807 

2 

4 

6 

9 

11 

43 

.6820 

.6833 

.6845 

.6858 

.6871 

6884 

6896 

.6909 

.6921 

.6934 

2 

4 

6 

8 

11 

44 

6947 

.6959 

.6972 

.6984 

.6997 

7009 

7022 

.7034 

.7046 

.7059 

2 

A 

6 

8 

10 

All  the  above  sines  are  less  than  1. 


NATURAL  SINES.    45°-90° 


143 


0 

0.0° 

0.1° 

0.2° 

0.3° 

0.4° 

0.5° 

0.6° 

0.7° 

0.8° 

0.9° 

-f-  Differences 

0' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48' 

54' 

1' 

2' 

3' 

4' 

5' 

45 

.7071 

.7083 

.7096 

.7108 

.7120 

.7133 

.7145 

.7157 

.7169 

.7181 

2 

4 

6 

8 

10 

46 

.7193 

.7206 

.7218 

.7230 

.7242 

.7254 

.7266 

.7278 

.7290 

.7302 

2 

4 

6 

8 

10 

47 

.7314 

.7325 

.7337 

.7349 

.7361 

.7373 

.7385 

.7396 

.7408 

.7420 

2 

4 

6 

8 

10 

48 

.7431 

.7443 

.7455 

.7466 

.7478 

.7490 

.7501 

.7513 

.7524 

.7536 

2 

4 

6 

8 

10 

49 

.7547 

.7559 

.7570 

.7581 

.7593 

.7604 

.7615 

.7627 

.7638 

.7649 

2 

4 

6 

8 

9 

50 

.7660 

.7672 

.7683 

.7694 

.7705 

.7716 

.7727 

.7738 

.7749 

.7760 

2 

4 

6 

7 

9 

51 

.7771 

.7782 

.7793 

.7804 

.7815 

.7826 

.7837 

.7848 

.7859 

.7869 

2 

4 

5 

7 

9 

52 

.7880 

.7891 

.7902 

.7912 

.7923 

.7934 

.7944 

.7955 

.7965 

.7976 

2 

4 

5 

7 

9 

53 

.7986 

.7997 

.8007 

.8018 

.8028 

.8039 

.8049 

.8059 

.8070 

.8080 

2 

3 

5 

7 

9 

54 

.8090 

.8100 

.8111 

.8121 

.8131 

.8141 

.8151 

.8161 

.8171 

.8181 

2 

3 

5 

7 

8 

55 

.8192 

.8202 

.8211 

.8221 

.8231 

.8241 

.8251 

.8261 

.8271 

.8281 

2 

3 

5 

7 

8 

56 

.8290 

.8300 

.8310 

.8320 

.8329 

.8339 

.8348 

.8358 

.8368 

.8377 

2 

3 

5 

6 

8 

57 

.8387 

.8396 

.8406 

.8415 

.8425 

.8434 

.8443 

.8453 

.8462 

.8471 

2 

3 

5 

6 

8 

58 

.8480 

.8490 

.8499 

.8508 

.8517 

.8526 

.8536 

.8545 

.8554 

.8563 

2 

3 

5 

6 

8 

59 

.8572 

.8581 

.8590 

.8599 

.8607 

.8616 

.8625 

.8634 

.8643 

.8652 

1 

3 

4 

6 

7 

60 

.8660 

.8669 

.8678 

.8686 

.8695 

.8704 

.8712 

.8721 

.8729 

.8738 

1 

3 

4 

6 

7 

61 

.8746 

.8755 

.8763 

.8771 

.8780 

.8788 

.8796 

.8805 

.8813 

.8821 

1 

3 

4 

6 

7 

62 

8829 

.8838 

.8846 

.8854 

.8862 

.8870 

.8878 

.8886 

.8894 

.8902 

1 

3 

4 

5 

7 

63 

8910 

.8918 

.8926 

.8934 

.8942 

.8949 

.8957 

.8965 

.8973 

.8980 

1 

3 

4 

5 

6 

64 

.8988 

.8996 

.9003 

.9011 

.9018 

.9026 

.9033 

.9041 

.9048 

.9056 

1 

3 

4 

5 

6 

65 

9063 

.9070 

.9078 

.9085 

.9092 

.9100 

.9107 

.9114 

.9121 

.9128 

1 

2 

4 

5 

6 

66 

9135 

.9143 

.9150 

.9157 

.9164 

.9171 

.9178 

.9184 

.9191 

.9198 

1 

2 

3 

5 

6 

67 

.9205 

.9212 

.9219 

.9225 

.9232 

.9239 

.9245 

.9252 

.9259 

.9265 

1 

2 

3 

4 

6 

68 

.9272 

.9278 

.9285 

.9291 

.9298 

.9304 

.9311 

.9317 

.9323 

.9330 

1 

2 

3 

4 

5 

69 

.9336 

.9342 

.9348 

.9354 

.9361 

.9367 

.9373 

.9379 

.9385 

.9391 

1 

2 

3 

4 

5 

70 

.9397 

.9403 

.9409 

.9415 

.9421 

.9426 

.9432 

.9438 

.9444 

.9449 

1 

2 

3 

4 

5 

71 

.9455 

.9461 

.9466 

.9472 

.9478 

.9483 

.9489 

.9494 

.9500 

.9505 

1 

2 

3 

4 

5 

72 

.9511 

.9516 

.9521 

.9527 

.9532 

.9537 

.9542 

.9548 

.9553 

.9558 

1 

2 

3 

4 

4 

73 

.9563 

.9568 

.9573 

.9578 

.9583 

.9588 

.9593 

.9598 

.9603 

.9608 

1 

2 

2 

3 

4 

74 

.9613 

.9617 

.9622 

.9627 

.9632 

.9636 

.9641 

.9646 

.9650 

.9655 

1 

2 

2 

3 

4 

75 

.9659 

.9664 

.9668 

.9673 

.9677 

.9681 

.9686 

.9690 

.9694 

.9699 

1 

1 

2 

3 

4 

76 

.9703 

.9707 

.9711 

.9715 

.9720 

.9724 

.9728 

.9732 

.9736 

.9740 

1 

1 

2 

3 

3 

77 

.9744 

.9748 

.9751 

.9755 

.9759 

.9763 

.9767 

.9770 

.9774 

.9778 

1 

2 

3 

3 

78 

.9781 

.9785 

.9789 

.9792 

.9796 

.9799 

.9803 

.9806 

.9810 

.9813 

1 

2 

2 

3 

79 

.9816 

.9820 

.9823 

.9826 

.9829 

.9833 

.9836 

.9839 

.9842 

.9845 

1 

2 

2 

3 

80 

.9848 

.9851 

.9854 

.9857 

.9860 

.9863 

.9866 

.9869 

.9871 

.9874 

0 

1 

2 

2 

81 

9877 

.9880 

.9882 

.9885 

.9888 

.9890 

.9893 

.9895 

.9898 

.9900 

0 

1 

2 

2 

82 

9903 

.9905 

.9907 

.9910 

.9912 

.9914 

.9917 

.9919 

.9921 

.9923 

0 

1 

2 

2 

83 

9925 

.9928 

.9930 

.9932 

.9934 

.9936 

.9938 

.9940 

.9942 

.9943 

0 

1 

1 

2 

84 

9945 

.9947 

.9949 

.9951 

.9952 

.9954 

.9956 

.9957 

.9959 

.9960 

0 

1 

1 

1 

85 

9962 

.9963 

.9965 

.9966 

.9968 

.9969 

.9971 

.9972 

.9973 

.9974 

0 

0 

1 

1 

1 

86 

9976 

.9977 

.9978 

.9979 

.9980 

.9981 

.9982 

.9983 

.9984 

.9985 

0 

0 

1 

1 

1 

87 

9986 

.9987 

.9988 

.9989 

.9990 

.9990 

.9991 

.9992 

.9993 

.9993 

0 

0 

0 

1 

1 

88 

9994 

.9995 

.9995 

.9996 

.9996 

.9997 

.9997 

.9997 

.9998 

.9998 

0 

0 

0 

0 

0 

89 

9998 

.9999 

.9999 

.9999 

.9999 

1.000 

1.000 

1.000 

1.000 

1.000 

0 

0 

0 

0 

0 

The  precise  value  of  all  sines  except  sin  90°  is  less  than  1. 


144 


NATUKAL  COSINES.    0°-45° 


o 

0.0° 

0.1° 

0.2° 

0.3° 

0.4° 

0.5° 

0.6° 

0.7° 

0.8° 

0.9° 

—  Differences 

0' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48' 

54' 

1' 

2' 

3' 

4' 

5' 

~0 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

.9999 

.9999 

.9999 

.9999 

0 

0 

0 

0 

0 

1 

.9998 

.9998 

.9998 

.9997 

.9997 

.9997 

.9996 

.9996 

.9995 

.9995 

0 

0 

0 

0 

0 

2 

.9994 

.9993 

.9993 

.9992 

.9991 

.9990 

.9990 

.9989 

.9988 

.9987 

0 

0 

0 

0 

0 

3 

.9986 

.9985 

.9984 

.9983 

.9982 

.9981 

.9980 

.9979 

.9978 

.9977 

0 

0 

1 

1 

1 

4 

.9976 

.9974 

.9973 

.9972 

.9971 

.9969 

.9968 

.9966 

.9965 

.9963 

0 

0 

1 

1 

1 

5 

.9962 

.9960 

.9959 

.9957 

.9956 

.9954 

.9952 

.9951 

.9949 

.9947 

0 

1 

1 

1 

1 

€ 

.9945 

.9943 

.9942 

.9940 

.9938 

.9936 

.9934 

.9932 

.9930 

.9928 

0 

1 

1 

1 

2 

7 

.9925 

.9923 

.9921 

.9919 

.9917 

.9914 

.9912 

.9910 

.9907 

.9905 

0 

1 

1 

2 

2 

8 

.9903 

.9900 

.9898 

.9895 

.9893 

.9890 

.9888 

.9885 

.9882 

.9880 

0 

1 

1 

2 

2 

9 

.9877 

.9874 

.9871 

.9869 

.9866 

.9863 

.9860 

.9857 

.9854 

.9851 

0 

1 

1 

2 

2 

10 

.9848 

.9845 

.9842 

.9839 

.9836 

.9833 

.9829 

.9826 

.9823 

.9820 

1 

1 

2 

2 

3 

11 

.9816 

.9813 

.9810 

.9806 

.9803 

.9799 

.9796 

.9792 

.9789 

.9785 

1 

1 

2 

2 

3 

12 

.9781 

.9778 

.9774 

.9770 

.9767 

.9763 

.9759 

.9755 

.9751 

.9748 

1 

1 

2 

3 

3 

13 

.9744 

.9740 

.9736 

.9732 

.9728 

.9724 

.9720 

.9715 

.9711 

.9707 

1 

1 

2 

3 

3 

14 

.9703 

.9699 

.9694 

.9690 

.9686 

.9681 

.9677 

.9673 

.9668 

.9664 

1 

1 

2 

3 

4 

15 

.9659 

.9655 

.9650 

.9646 

.9641 

.9636 

.9632 

.9627 

.9622 

.9617 

1 

2 

2 

3 

4 

16 

.9613 

.9608 

.9603 

.9598 

.9593 

.9588 

.9583 

9578 

.9573 

.9568 

1 

2 

2 

3 

4 

17 

.9563 

.9558 

.9553 

.9548 

.9542 

.9537 

.9532 

9527 

.9521 

.9516 

1 

2 

3 

4 

4 

18 

.9511 

.9505 

.9500 

.9494 

.9489 

.9483 

.9478 

9472 

.9466 

.9461 

1 

2 

3 

4 

5 

19 

.9455 

.9449 

.9444 

.9438 

.9432 

.9426 

.9421 

9415 

.9409 

.9403 

1 

2 

3 

4 

5 

20 

.9397 

.9391 

.9385 

.9379 

.9373 

.9367 

.9361 

9354 

.9348 

.9342 

1 

2 

3 

4 

5 

21 

.9336 

.9330 

.9323 

.9317 

.9311 

.9304 

.9298 

9291 

.9285 

.9278 

1 

2 

3 

4 

5 

22 

.9272 

.9265 

.9259 

.9252 

.9245 

9239 

.9232 

9225 

.9219 

.9212 

1 

2 

3 

4 

6 

23 

9205 

.9198 

.9191 

9184 

.9178 

9171 

.9164 

9157 

.9150 

.9143 

1 

2 

3 

5 

6 

24 

9135 

9128 

.9121 

.9114 

.9107 

.9100 

.9092 

9085 

.9078 

.9070 

1 

2 

4 

5 

6 

25 

9063 

9056 

9048 

9041 

9033 

9026 

.9018 

9011 

9003 

8996 

1 

3 

4 

5 

6 

26 

8988 

8980 

8973 

8965 

8957 

8949 

.8942 

8934 

8926 

8918 

1 

3 

4 

5 

6 

27 

8910 

8902 

8894 

8886 

8878 

8870 

8862 

8854 

8846 

8838 

1 

3 

4 

5 

7 

28 

8829 

8821 

8813 

8805 

8796 

8788 

8780 

8771 

8763 

8755 

1 

3 

4 

6 

7 

29 

8746 

8738 

8729 

8721 

8712 

8704 

8695 

8686 

8678 

8669 

1 

3 

4 

6 

7 

30 

8660 

8652 

8643 

8634 

8625 

8616 

8607 

8599 

8590 

8581 

1 

3 

4 

6 

7 

31 

8572 

8563 

8554 

8545 

8536 

8526 

8517 

8508 

8499 

8490 

2 

3 

5 

6 

8 

32 

8480 

8471 

8462 

8453 

8443 

8434 

8425 

8415 

8406 

8396 

2 

3 

5 

6 

8 

33 

8387 

8377 

8368 

8358 

8348 

8339 

8329 

8320 

8310 

8300 

2 

3 

5 

6 

8 

34 

8290 

8281 

8271 

8261 

8251 

8241 

8231 

8221 

8211 

8202 

2 

3 

5 

7 

8 

35 

8192 

8181 

8171 

8161 

8151 

8141 

8131 

8121 

8111 

8100 

2 

3 

5 

7 

8 

36 

8090 

8080 

8070 

8059 

8049 

8039 

8028 

8018 

8007 

7997 

2 

3 

5 

7 

9 

37 

7986 

7976 

7965 

7955 

7944 

7934 

7923 

7912 

7902 

7891 

2 

4 

5 

7 

9 

38 

7880 

7869 

7859 

7848 

7837 

7826 

7815 

7804 

7793 

7782 

2 

4 

5 

7 

9 

39 

7771 

7760 

7749 

7738 

7727 

7716 

7705 

7694 

7683 

7672 

2 

4 

6 

7 

9 

40 

7660 

7649 

7638 

7627 

7615 

7604 

7593 

7581 

7570 

7559 

2 

4 

6 

8 

9 

41 

7547 

7536 

7524 

7513 

7501 

7490 

7478 

7466 

7455 

7443 

2 

4 

6 

8 

10 

42 

7431 

7420 

7408 

7396 

7385 

7373 

7361 

7349 

7337 

7325 

2 

4 

6 

8 

10 

43 

7314 

7302 

7290 

7278 

7266 

7254 

7242 

7230 

7218 

7206 

2 

4 

6 

8 

10 

44 

7193 

7181 

7169 

7157 

7145 

7133 

7120 

7108 

7096 

7083 

2 

4 

6 

8 

10 

The  precise  value  of  all  cosines  except  cos  0°  is  less  than  1. 


NATURAL  COSINES.    45°-90° 


145 


0 

0.0° 

0.1° 

0.2° 

0.3° 

0.4° 

0.5° 

0.6° 

0.7° 

0.8° 

0.9° 

—  Differences 

0' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48' 

54' 

1' 

2' 

3' 

4' 

5' 

"45 

.7071 

.7059 

.7046 

.7034 

.7022 

.7009 

.6997 

.6984 

.6972 

.69591  2 

4 

6 

8 

10 

46 

.6947 

.6934 

.6921 

.6909 

.6896 

.6884 

.6871 

.6858 

.6845 

.6833  2 

4 

6 

8 

11 

47 

.6820 

.6807 

.6794 

.6782 

.6769 

.6756 

.6743 

.6730 

.6717 

.6704 

2 

4 

6 

9 

11 

48 

.6691 

.6678 

.6665 

.6652 

.6639 

.6626 

.6613 

.6600 

.6587 

.6574 

2 

4 

7 

9 

11 

49 

.6561 

.6547 

.6534 

.6521 

.6508 

.6494 

.6481 

.6468 

.6455 

.6441 

2 

4 

7 

9 

11 

50 

.6428 

.6414 

.6401 

.6388 

.6374 

.6361 

.6347 

.6334 

.6320 

.6307 

2 

4 

7 

9 

11 

51 

.6293 

.6280 

.6266 

.6252 

.6239 

.6225 

.6211 

.6198 

.6184 

.6170 

2 

5 

7 

9 

11 

52 

.6157 

.6143 

.6129 

.6115 

.6101 

.6088 

.6074 

.6060 

.6046 

.6032 

2 

5 

7 

9 

12 

53 

.6018 

.6004 

.5990 

.5976 

.5962 

.5948 

.5934 

.5920 

.5906 

.5892 

2 

5 

7 

9 

12 

54 

.5878 

.5864 

.5850 

.5835 

.5821 

.5807 

.5793 

.5779 

.5764 

.5750 

2 

5 

7 

9 

12 

55 

.5736 

.5721 

.5707 

.5693 

.5678 

.5664 

.5650 

.5635 

.5621 

.5606 

2 

5 

7 

10 

12 

56 

5592 

.5577 

.5563 

.5548 

.5534 

.5519 

.5505 

.5490 

.5476 

.5461 

2 

5 

7 

10 

12 

57 

5446 

.5432 

.5417 

.5402 

.5388 

.5373 

.5358 

.5344 

.5329 

.5314 

2 

5 

7 

10 

12 

58 

5299 

.5284 

.5270 

.5255 

.5240 

.5225 

.5210 

.5195 

.5180 

.5165 

2 

5 

7 

10 

12 

59 

5150 

.5135 

.5120 

.5105 

.5090 

.5075 

.5060 

.5045 

.5030 

.5015 

3 

5 

8 

10 

13 

60 

5000 

.4985 

.4970 

.4955 

.4939 

.4924 

.4909 

.4894 

.4879 

.4863 

3 

5 

8 

10 

13 

61 

4848 

.4833 

.4818 

.4802 

.4787 

.4772 

.4756 

.4741 

.4726 

.4710 

3 

5 

8 

10 

13 

62 

4695 

.4679 

.4664 

.4648 

.4633 

.4617 

.4602 

.4586 

.4571 

.4555 

3 

5 

8 

10 

13 

63 

4540 

4524 

.4509 

.4493 

.4478 

.4462 

.4446 

.4431 

.4415 

.4399 

3 

5 

8 

10 

13 

64 

4384 

4368 

.4352 

.4337 

.4321 

.4305 

.4289 

.4274 

.4258 

.4242 

3 

5 

8 

11 

13 

65 

4226 

4210 

.4195 

.4179 

.4163 

.4147 

.4131 

.4115 

.4099 

.4083 

3 

5 

8 

11 

13 

66 

4067 

4051 

.4035 

.4019 

.4003 

.3987 

.3971 

.3955 

.3939 

.3923 

3 

5 

8 

11 

13 

67 

3907 

3891 

.3875 

.3859 

.3843 

.3827 

.3811 

.3795 

.3778 

.3762 

3 

5 

8 

11 

13 

68 

3746 

3730 

.3714 

.3697 

.3681 

.3665 

.3649 

.3633 

.3616 

.3600 

3 

5 

8 

11 

14 

69 

3584 

3567 

.3551 

.3535 

.3518 

.3502 

.3486 

.3469 

.3453 

.3437 

3 

5 

8 

11 

14 

70 

3420 

3404 

.3387 

.3371 

.3355 

.3338 

.3322 

.3305 

.3289 

.3272  3 

5 

8 

11 

14 

71 

3256 

3239 

.3223 

.3206 

.3190 

.3173 

.3156 

.3140 

.3123 

.3107  3 

6 

8 

11 

14 

72 

3090 

3074 

.3057 

.3040 

.3024 

.3007 

.2990 

.2974 

.2957 

.2940  3 

6 

8 

11 

14 

73 

2924 

2907 

2890 

.2874 

.2857 

2840 

.2823 

.2807 

.2790 

.2773  3 

6 

8 

11 

14 

74 

2756 

2740 

2723 

.2706 

.2689 

2672 

.2656 

.2639 

.2622 

.2605  3 

6 

8 

11 

14 

75 

2588 

2571 

2554 

.2538 

.2521 

2504 

.2487 

.2470 

.2453 

.2436  3 

6 

8 

11 

14 

76 

2419 

2402 

2385 

.2368 

.2351 

2334 

.2317 

.2300 

.2284 

.2267  3 

6 

8 

11 

14 

77 

2250 

2233 

2215 

.2198 

.2181 

2164 

.2147 

.2130 

.2113 

.2096  3 

6 

9 

11 

14 

78 

2079 

2062 

2045 

.2028 

.2011 

1994 

.1977 

.1959 

.1942 

.1925 

3 

6 

9 

11 

14 

79 

1908 

1891 

.1874 

.1857 

.1840 

1822 

.1805 

.1788 

.1771 

.1754 

3 

6 

9 

11 

14 

80 

1736 

.1719 

.1702 

.1685 

.1668 

1650 

.1633 

.1616 

.1599 

.1582 

3 

6 

9 

11 

14 

81 

1564 

.1547 

.1530 

.1513 

.1495 

1478 

.1461 

.1444 

.1426 

.1409  3 

6 

9 

12 

14 

82 

1392 

1374 

.1357 

.1340 

.1323 

1305 

.1288 

.1271 

.1253 

.1236  3 

6 

9 

12 

14 

83 

1219 

1201 

1184 

.1167 

.1149 

1132 

.1115 

.1097 

.1080 

.1063 

3 

6 

9 

12 

14 

84 

1045 

1028 

1011 

.0993 

.0976 

0958 

0941 

.0924 

.0906 

.0889 

3 

6 

9 

12 

14 

85 

0872 

0854 

0837 

.0819 

.0802 

0785 

0767 

.0750 

.0732 

.0715 

3 

6 

9 

12 

14 

86 

0698 

0680 

0663 

.0645 

.0628 

0610 

0593 

0576 

0558 

0541 

3 

6 

9 

12 

15 

87 

0523 

0506 

0488 

0471 

.0454 

0436 

0419 

0401 

0384 

0366 

3 

6 

9 

12 

15 

88 

0349 

0332 

0314 

0297 

.0279 

0262 

0244 

0227 

0209 

0192  3 

6 

9 

12 

15 

89 

0175 

0157 

0140 

0122 

.0105 

0087 

0070 

0052 

0035 

0017  3 

6 

9 

12 

15 

All  the  above  cosines  are  less  than  1. 


146 


NATURAL  TA^GEXTS.    0°-45e 


o 

0.0° 

0.1° 

0.2° 

0.3° 

0.4° 

0.5° 

0.6° 

0.7° 

0.8° 

0.9° 

+  Differences 

0' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48' 

54' 

1' 

2' 

3' 

4' 

5' 

0 

0.0000 

.0017 

.0035 

.0052 

.0070 

.0087 

.0105 

.0122 

.0140 

.0157 

3 

6 

9 

12 

15 

1 

0.0175 

.0192 

.0209 

.0227 

.0244 

.0262 

.0279 

.0297 

.0314 

.0332 

3 

6 

9 

12 

15 

2 

0.0349 

.0367 

.0384 

.0402 

.0419 

.0437 

.0454 

.0472 

.0489 

.0507 

3 

6 

9 

12 

15 

3 

0.0524 

.0542 

.0559 

.0577 

.0594 

.0612 

.0629 

.0647 

.0664 

.0682 

3 

6 

9 

12 

15 

4 

0.0699 

.0717 

.0734 

.0752 

.0769 

.0787 

.0805 

.0822 

.0840 

.0857 

3 

6 

9 

12 

15 

5 

0.0875 

.0892 

.0910 

.0928 

.0945 

.0963 

.0981 

.0998 

.1016 

.1033 

3 

6 

9 

12 

15 

6 

0.1051 

.1069 

.1086 

.1104 

.1122 

.1139 

.1157 

.1175 

.1192 

.1210 

3 

6 

9 

12 

15 

7 

0.1228 

.1246 

.1263 

.1281 

.1299 

.1317 

.1334 

.1352 

.1370 

.1388 

3 

6 

9 

12 

15 

8 

0.1405 

.1423 

.1441 

.1459 

.1477 

.1495 

.1512 

.1530 

.1548 

.1566 

3 

6 

9 

12 

15 

9 

0.1584 

.1602 

.1620 

.1638 

.1655 

.1673 

.1691 

.1709 

.1727 

.1745 

3 

6 

9 

12 

15 

10 

0.1763 

.1781 

.1799 

.1817 

.1835 

.1853 

.1871 

.1890 

.1908 

.1926 

3 

6 

9 

12 

15 

11 

0.1944 

.1962 

.1980 

.1998 

.2016 

.2035 

.2053 

.2071 

.2089 

.2107 

3 

6 

9 

12 

15 

12 

0.2126 

.2144 

.2162 

.2180 

.2199 

.2217 

.2235 

.2254 

.2272 

.2290 

3 

6 

9 

12 

15 

13 

0.2309 

.2327 

.2345 

.2364 

.2382 

.2401 

.2419 

.2438 

.2456 

.2475 

3 

6 

9 

12 

15 

14 

0.2493 

.2512 

.2530 

.2549 

.2568 

.2586 

.2605 

.2623 

.2642 

.2661 

3 

6 

9 

12 

16 

15 

0.2679 

.2698 

.2717 

.2736 

.2754 

2773 

.2792 

.2811 

.2830 

.2849 

3 

6 

9 

13 

16 

16 

0.2867 

.2886 

.2905 

.2924 

.2943 

.2962 

.2981 

.3000 

.3019 

.3038 

3 

6 

9 

13 

16 

17 

0.3057 

.3076 

.3096 

.3115 

.3134 

.3153 

.3172 

.3191 

.3211 

.3230 

3 

6 

10 

13 

16 

18 

0.3249 

.3269 

.3288 

.3307 

.3327 

.3346 

.3365 

.3385 

.3404 

.3424 

3 

6 

10 

13 

16 

19 

0.3443 

.3463 

.3482 

.3502 

.3522 

3541 

.3561 

.3581 

.3600 

.3620 

3 

7 

10 

13 

16 

20 

0.3640 

.3659 

.3679 

.3699 

.3719 

3739 

.3759 

.3779 

.3799 

.3819 

3 

7 

10 

13 

17 

21 

0.3839 

.3859 

.3879 

.3899 

.3919 

3939 

.3959 

.3979 

.4000 

.4020 

3 

7 

10 

13 

17 

22 

0.4040 

.4061 

.4081 

.4101 

.4122 

4142 

.4163 

.4183 

.4204 

.4224 

3 

7 

10 

14 

17 

23 

0.4245 

.4265 

.4286 

.4307 

.4327 

434C 

.4369 

.4390 

.4411 

.4431 

3 

7 

10 

14 

17 

24 

0.4452 

.4473 

.4494 

.4515 

.4536 

4557 

.4578 

.4599 

.4621 

.4642 

4 

7 

11 

14 

18 

25 

0.4663 

.4684 

.4706 

.4727 

.4748 

4770 

.4791 

.4813 

.4834 

.4856 

4 

7 

11 

14 

18 

20 

0.4877 

.4899 

.4921 

.4942 

.4964 

4986 

.5008 

.5029 

.5051 

.5073 

4 

7 

11 

15 

18 

27 

0.5095 

.5117 

.5139 

.5161 

.5184 

5206 

.5228 

.5250 

.5272 

.5295 

4 

7 

11 

15 

18 

28 

0.5317 

.5340 

.5362 

.5384 

.5407 

5430 

.5452 

.5475 

.5498 

.5520 

4 

8 

11 

15 

19 

29 

0.5543 

.5566 

.5589 

.5612 

.5635 

5658 

.5681 

.5704 

.5727 

.5750 

4 

8 

12 

15 

19 

30 

0.5774 

.5797 

.5820 

.5844 

.5867 

5890 

.5914 

.5938 

.5961 

.5985 

4 

8 

12 

16 

20 

31 

0.6009 

.6032 

.6056 

.6080 

.6104 

6128 

.6152 

.6176 

.6200 

.6224 

4 

8 

12 

16 

20 

32 

0.6249 

.6273 

.6297 

.6322 

.6346 

.6371 

.6395 

.6420 

.6445 

.6469 

4 

8 

12 

16 

20 

33 

0.6494 

.6519 

.6544 

.6569 

.6594 

6619 

.6644 

.6669 

.6694 

.6720 

4 

8 

13 

17 

21 

34 

0.6745 

.6771 

.6796 

.6822 

.6847 

6873 

.6899 

.6924 

.6950 

.6976 

4 

9 

13 

17 

21 

35 

0.7002 

.7028 

.7054 

.7080 

.7107 

7133 

.7159 

.7186 

.7212 

.7239 

4 

9 

13 

18 

22 

36 

0.7265 

.7292 

.7319 

.7346 

.7373 

7400 

.7427 

.7454 

.7481 

.7508 

5 

9 

14 

18 

22 

37 

0.7536 

.7563 

.7590 

.7618 

.7646 

7673 

.7701 

.7729 

.7757 

.7785 

5 

9 

14 

18 

23 

38 

0.7813 

.7841 

.7869 

.7898 

.7926 

.7954 

.7983 

.8012 

.8040 

.8069 

5 

9 

14 

19 

24 

39 

0.8098 

.8127 

.8156 

.8185 

.8214 

8243 

.8273 

.8302 

.8332 

.8361 

5 

10 

15 

20 

24 

40 

0.8391 

.8421 

.8451 

.8481 

.8511 

.8541 

.8571 

.8601 

.8632 

.8662 

5 

10 

15 

20 

25 

41 

0.8693 

.8724 

.8754 

.8785 

.8816 

8847 

.8878 

.8910 

.8941 

.8972 

5 

10 

16 

21 

26 

42 

0.9004 

.9036 

.9067 

.9099 

.9131 

.9163 

.9195 

.9228 

.9260 

.9293 

5 

11 

16 

21 

27 

43 

0.9325 

.9358 

.9391 

.9424 

.9457 

9490 

.9523 

.9556 

.9590 

.9623 

6 

11 

17 

22 

28 

44 

0.9657 

.9691 

.9725 

.9759 

.9793 

.9827 

.9861 

.9896 

.9930 

.9965 

6 

11 

17 

23 

29 

All  tangents  less  than  tan  45°  are  less  than  1. 


NATURAL  TANGENTS.    45°- 90° 


147 


0 

0.0° 

0.1° 

0.2° 

0.3° 

0.4° 

0.5° 

0.6° 

0.7° 

0.8° 

0.9° 

+  Differences 

0' 

6' 

12' 

IB' 

24' 

30' 

36' 

42' 

48' 

54' 

1' 

2' 

3' 

4' 

5' 

45 

1.0000 

.0035 

.0070 

.0105 

.0141 

.0176 

.0212 

.0247 

.0283 

.0319 

6 

12 

18 

~24 

30 

46 

1.0355 

.0392 

.0428 

.0464 

.0501 

.0538 

.0575 

.0612 

.0649 

.0686 

6 

12 

18 

25 

31 

47 

1.0724 

.0761 

.0799 

.0837 

.0875 

.0913 

.0951 

.0990 

.1028 

.1067 

6 

13 

19 

25 

32 

48 

1.1106 

.1145 

.1184 

.1224 

.1263 

.1303 

.1343 

.1383 

.1423 

.1463 

7 

13 

20 

26 

33 

49 

1.1504 

.1544 

.1585 

.1626 

.1667 

.1708 

.1750 

.1792 

.1833 

.1875 

7 

14 

21 

28 

34 

50 

1.1918 

.1960 

.2002 

.2045 

.2088 

.2131 

.2174 

.2218 

.2261 

.2305 

7 

14 

22 

29 

36 

51 

1.2349 

.2393 

.2437 

.2482 

.2527 

.2572 

.2617 

.2662 

.2708 

.2753 

8 

15 

23 

30 

38 

52 

1.2799 

.2846 

.2892 

.2938 

.2985 

.3032 

.3079 

.3127 

.3175 

.3222 

8 

16 

24 

31 

39 

53 

1.3270 

.3319 

.3367 

.3416 

.3465 

.3514 

.3564 

.3613 

.3663 

.3713 

8 

16 

25 

33 

41 

54 

1.3764 

.3814 

.3865 

.3916 

.3968 

.4019 

.4071 

.4124 

.4176 

.4229 

9 

17 

26 

34 

43 

55 

1.4281 

.4335 

.4388 

.4442 

.4496 

.4550 

.4605 

.4659 

.4715 

.4770 

9 

18 

27 

36 

45 

56 

1.4826 

.4882 

.4938 

.4994 

.5051 

.5108 

.5166 

.5224 

.5282 

.5340 

10 

19 

29 

38 

48 

57 

1.5399 

.5458 

.5517 

.5577 

.5637 

.5697 

.5757 

.5818 

.5880 

.5941 

10 

20 

30 

40 

50 

58 

1.6003 

6066 

.6128 

.6191 

.6255 

.6319 

.6383 

.6447 

.6512 

.6577 

11 

21 

32 

43 

53 

59 

1.6643 

6709 

.6775 

.6842 

.6909 

.6977 

.7045 

.7113 

.7182 

.7251 

11 

23 

34 

45 

56 

60 

1.7321 

7391 

.7461 

.7532 

.7603 

.7675 

.7747 

.7820 

.7893 

.7966 

12 

24 

36 

48 

60 

61 

1.8040 

8115 

.8190 

.8265 

.8341 

.8418 

.8495 

.8572 

.8650 

.8728 

13 

26 

38 

51 

64 

62 

1.8807 

8887 

.8967 

.9047 

.9128 

.9210 

.9292 

.9375 

.9458 

.9542 

14 

27 

41 

55 

68 

63 

1.9626 

9711 

.9797 

.9883 

.9970 

.0057 

.0145 

.0233 

.0323 

.0413 

15 

29 

44 

58 

73 

64 

2.0503 

0594 

.0686 

.0778 

.0872 

.0965 

.1060 

.1155 

.1251 

.1348 

16 

31 

47 

63 

78 

65 

2.1445 

1543 

.1642 

.1742 

.1842 

.1943 

.2045 

.2148 

.2251 

.2355 

17 

34 

51 

68 

85 

66 

2.2460 

2566 

.2673 

.2781 

.2889 

.2998 

.3109 

.3220 

.3332 

.3445 

18 

37 

55 

73 

92 

67 

2.3559 

3673 

.3789 

.3906 

.4023 

.4142 

.4262 

.4383 

.4504 

.4627 

20 

40 

60 

79 

99 

68 

2.4751 

4876 

.5002 

.5129 

.5257 

.5386 

.5517 

.5649 

.5782 

.5916 

22 

43 

65 

87 

108 

69 

2.6051 

6187 

.6325 

.6464 

.6605 

.6746 

.6889 

.7034 

.7179 

.7326 

24 

47 

71 

95 

119 

70 

2.7475 

7625 

.7776 

.7929 

.8083 

.8239 

.8397 

.8556 

.8716 

.8878 

26 

52 

78 

104 

130 

71 

2.9042 

9208 

.9375 

.9544 

.9714 

.9887 

.0061 

.0237 

.0415 

.0595 

29 

58 

87 

116 

144 

72 

3.0777 

0961 

.1146 

.1334 

.1524 

.1716 

.1910 

.2106 

.2305 

.2506 

32 

64 

96 

129 

161 

73 

3.2709 

2914 

.3122 

.3332 

.3544 

.3759 

.3977 

.4197 

.4420 

.4646 

36 

72 

108 

144 

180 

74 

3.4874 

5105 

.5339 

.5576 

.5816 

.6059 

.6305 

.6554 

.6806 

.7062 

41 

81 

122 

1631204 

75 

3.7321 

7583 

.7848 

.8118 

.8391 

.8667 

.8947 

.9232 

.9520 

.9812 

76 

4.0108 

0408 

.0713 

.1022 

.1335 

.1653 

.1976 

.2303 

.2635 

.2972 

77 

4.3315 

3662 

.4015 

.4373 

.4737 

.5107 

.5483 

.5864 

.6252 

.6646 

78 

4.7046 

7453 

.7867 

.8288 

.8716 

.9152 

.9594 

.0045 

.0504 

.0970 

Use  ordinary 

79 

5.1446 

1929 

.2422 

.2924 

.3435 

.3955 

.4486 

.5026 

.5578 

.6140 

interpolation. 

80 

5.6713 

7297 

.7894 

.8502 

.9124 

.9758 

.0405 

.1066 

.1742 

2432 

81 

6.3138 

3859 

.4596 

.5350 

.6122 

.6912 

.7720 

.8548 

.9395 

0264 

82 

7.1154 

.2066 

.3002 

.3962 

.4947 

.5958 

.6996 

.8062 

.9158 

0285 

83 

8.1443 

2636 

.3863 

.5126 

.6427 

.7769 

.9152 

.0579 

.2052 

3572 

84 

9.5144 

6768 

.8448 

.0187 

.1988 

.3854 

.5789 

.7797 

.9882 

2048 

85 

11.430 

11.66 

11.91 

12.16 

12.43 

12.71 

13.00 

13.30 

13.62 

13.95 

86 

14.301 

14.67 

15.06 

15.46 

15.89 

16.35 

16.83 

17.34 

17.89 

18.46 

87 

19.081 

19.74 

20.45 

21.20 

22.02 

22.90 

23.86 

24.90 

26.03 

27.27 

88 

28.636 

30.14 

31.82 

33.69 

35.80 

38.19 

40.92 

44.07 

47.74 

52.08 

89 

57.290 

63.66 

71.62 

81.85 

95.49 

114.6 

143.2 

191.0 

286.5 

573.0 

The  integral  part  of  tangents  in  heavy-face  type  is  1  greater  than  preceding  part. 


148 


NATUKAL  COTANGENTS.    0°-45° 


0 

0.0° 

0.1° 

0.2° 

0.3° 

0.4° 

0.5° 

0.6° 

0.7° 

0.8° 

0.9° 

—  Differences 

0' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48' 

54' 

1' 

2'|  3' 

4' 

5' 

0 

00 

573.0 

286.5 

191.0 

143.2 

114.6 

95.49 

81.85 

71.62 

63.66 

1 

57.290 

52.08 

47.74 

44.07 

40.92 

38.19 

35.80 

33.69 

31.82 

30.14 

2 

28.636 

27.27 

26.03 

24.90 

23.86 

22.90 

22.02 

21.20 

20.45 

19.74 

3 
4 

19.081 
14.301 

18.46 
13.95 

17.89 
13.62 

17.34 
13.30 

16.83 
13.00 

16.35 
12.71 

15.89 
12.43 

15.46 
12.16 

15.06 
11.91 

14.67 
11.66 

Use  ordinary 

interpolation. 

5 

11.430 

.2048 

.9882 

.7797 

.5789 

.3854 

.1988 

.0187 

.8448 

.6768 

6 

9.5144 

.3572 

.2052 

.0579 

.9152 

.7769 

.6427 

.5126 

.3863 

.2636 

7 

8.1443 

.0285 

.9158 

.8062 

.6996 

.5958 

.4947 

.3962 

.3002 

.2066 

8 

7.1154 

.0264 

.9395 

.8548 

.7720 

.6912 

.6122 

.5350 

.4596 

.3859 

9 

6.3138 

.2432 

.1742 

.1066 

.0405 

.9758 

.9124 

.8502 

.7894 

.7297 

10 

5.6713 

.6140 

.5578 

.5026 

.4486 

.3955 

.3435 

.2924 

.2422 

.1929 

11 

5.1446 

.0970 

.0504 

.0045 

.9594 

.9152 

.8716 

.8288 

.7867 

.7453 

12 

4.7046 

.6646 

.6252 

.5864 

.5483 

.5107 

.4737 

.4373 

.4015 

.3662 

13 

4.3315 

.2972 

.2635 

.2303 

.1976 

.1653 

.1335 

.1022 

.0713 

.0408 

14 

4.0108 

.9812 

.9520 

.9232 

.8947 

.8667 

.8391 

.8118 

.7848 

.7583 

15 

3.7321 

.7062 

.6806 

.6554 

.6305 

.6059 

.5816 

.5576 

.5339 

.5105 

41 

81 

122 

163 

203 

16 

3.4874 

.4646 

.4420 

.4197 

.3977 

.3759 

.3544 

.3332 

.3122 

.2914 

36 

72 

108 

144 

180 

17 

3.2709 

.2506 

.2305 

.2106 

.1910 

.1716 

.1524 

.1334 

.1146 

.0961 

32 

64 

96 

129 

161 

18 

3.0777 

.0595 

.0415 

.0237 

.0061 

.9887 

.9714 

.9544 

.9375 

.9208 

29 

58 

87 

116 

144 

19 

2.9042 

.8878 

.8716 

.8556 

.8397 

.8239 

.8083 

.7929 

.7776 

.7625 

26 

52 

78 

104 

130 

20 

2.7475 

.7326 

.7179 

.7034 

.6889 

.6746 

.6605 

.6464 

.6325 

.6187 

24 

47 

71 

95 

119 

21 

2.6051 

.5916 

.5782 

.5649 

.5517 

.5386 

.5257 

.5129 

.5002 

.4876 

22 

43 

65 

87 

108 

22 

2.4751 

.4627 

.4504 

.4383 

.4262 

.4142 

.4023 

.3906 

.3789 

.3673 

20 

40 

60 

79 

99 

23 

2.3559 

.3445 

.3332 

.3220 

.3109 

.2998 

.2889 

.2781 

.2673 

.2566 

18 

37 

55 

73 

92 

24 

2.2460 

.2355 

.2251 

.2148 

.2045 

.1943 

.1842 

.1742 

.1642 

.1543 

17 

34 

51 

68 

85 

25 

2.1445 

.1348 

.1251 

.1155 

.1060 

.0965 

.0872 

.0778 

.0686 

.0594 

16 

31 

47 

63 

78 

26 

2.0503 

.0413 

.0323 

.0233 

.0145 

.0057 

.9970 

.9883 

.9797 

.9711 

15 

29 

44 

58 

73 

27 

1.9626 

.9542 

.9458 

.9375 

.9292 

.9210 

.9128 

.9047 

.8967 

.8887 

14 

27 

41 

55 

68 

28 

1.8807 

.8728 

.8650 

.8572 

.8495 

.8418 

.8341 

.8265 

.8190 

.8115 

13 

26 

38 

51 

64 

29 

1.8040 

.7966 

.7893 

.7820 

.7747 

.7675 

.7603 

.7532 

.7461 

.7391 

12 

24 

36 

48 

60 

30 

1.7321 

.7251 

.7182 

.7113 

.7045 

.6977 

.6909 

.6842 

.6775 

.6709 

11 

23 

34 

45 

56 

31 

1.6643 

.6577 

.6512 

.6447 

.6383 

.6319 

.6255 

.6191 

.6128 

.6066 

11 

21 

32 

43 

53 

32 

1.6003 

.5941 

.5880 

.5818 

.5757 

.5697 

.5637 

.5577 

.5517 

.5458 

10 

20 

30 

40 

50 

33 

1.5399 

.5340 

.5282 

.5224 

.5166 

.5108 

.5051 

.4994 

.4938 

.4882 

10 

19 

29 

38 

48 

34 

1.4826 

.4770 

.4715 

.4659 

.4605 

.4550 

.4496 

.4442 

.4388 

.4335 

9 

18 

27 

36 

45 

35 

1.4281 

.4229 

.4176 

.4124 

.4071 

.4019 

.3968 

.3916 

.3865 

.3814 

9 

17 

26 

34 

43 

36 

1.3764 

.3713 

.3663 

.3613 

.3564 

.3514 

.3465 

.3416 

.3367 

.3319 

8 

16 

25 

33 

41 

37 

1.3270 

.3222 

.3175 

.3127 

.3079 

.3032 

.2985 

.2938 

.2892 

.2846 

8 

16 

24 

31 

39 

38 

1.2799 

.2753 

.2708 

.2662 

.2617 

.2572 

.2527 

.2482 

.2437 

.2393 

8 

15 

23 

30 

38 

39 

1.2349 

.2305 

.2261 

.2218 

.2174 

.2131 

.2088 

.2045 

.2002 

.1960 

7 

14 

22 

29 

36 

40 

1.1918 

.1875 

.1833 

.1792 

.1750 

.1708 

.1667 

.1626 

.1585 

.1544 

7 

14 

21 

28 

34 

41 

1.1504 

.1463 

.1423 

.1383 

.1343 

.1303 

.1263 

.1224 

.1184 

.1145 

7 

13 

20 

26 

33 

42 

1.1106 

.1067 

.1028 

.0990 

.0951 

.0913 

.0875 

.0837 

.0799 

.0761 

6 

13 

19 

25 

32 

43 

1.0724 

.0686 

.0649 

.0612 

.0575 

.0538 

.0501 

.0464 

.0428 

.0392 

6 

12 

18 

25 

31 

44 

1.0355 

.0319 

.0283 

.0247 

.0212 

.0176 

.0141 

.0105 

.0070 

.0035 

6 

12 

18 

24 

30 

The  integral  part  of  cotangents  in  heavy-face  type  is  1  less  than  preceding  part. 


NATUKAL  COTANGENTS.    45°-90° 


149 


o 

0.0° 

0.1° 

0.2° 

0.3° 

0.4° 

0.5° 

0.6° 

0.7° 

0.8° 

0.9° 

—  Differences 

0' 

6' 

12' 

18' 

24' 

30' 

36' 

42' 

48' 

54' 

1' 

2' 

3' 

4' 

5' 

"45 

1.0000 

9965 

9930 

9896 

9861 

9827 

9793 

9759 

9725 

9691 

5 

11 

17 

23 

29 

46 

0.9657 

9623 

9590 

9556 

9523 

9490 

9457 

9424 

9391 

9358 

5 

11 

17 

22 

28 

47 

0.9325 

9293 

9260 

9228 

9195 

9163 

9131 

9099 

9067 

9036 

5 

11 

16 

21 

27 

48 

0.9004 

8972 

8941 

8910 

8878 

8847 

8816 

8785 

8754 

8724 

5 

10 

16 

21 

26 

49 

0.8693 

8662 

8632 

8601 

8571 

8541 

8511 

8481 

8451 

8421 

5 

10 

15 

20 

25 

50 

0.8391 

8361 

8332 

8302 

8273 

8243 

8214 

8185 

8156 

8127 

5 

10 

15 

20 

24 

51 

0.8098 

8069 

8040 

8012 

7983 

7954 

7926 

7898 

7869 

7841 

5 

9 

14 

19 

24 

52 

0.7813 

7785 

7757 

7729 

7701 

7673 

7646 

7618 

7590 

7563 

5 

9 

14 

18 

23 

53 

0.7536 

7508 

7481 

7454 

7427 

7400 

7373 

7346 

7319 

7292 

5 

9 

14 

18 

23 

54 

0.7265 

7239 

7212 

7186 

7159 

7133 

7107 

7080 

7054 

7028 

4 

9 

13 

18 

22 

55 

0.7002 

6976 

6950 

6924 

6899 

6873 

6847 

6822 

6796 

6771 

4 

9 

13 

17 

21 

56 

0.6745 

6720 

6694 

6669 

6644 

6619 

6594 

6569 

6544 

6519 

4 

8 

13 

17 

21 

57 

0.6494 

6469 

6445 

6420 

6395 

6371 

6346 

6322 

6297 

6273 

4 

8 

12 

16 

20 

58 

0.6249 

6224 

6200 

6176 

6152 

6128 

6104 

6080 

6056 

6032 

4 

8 

12 

16 

20 

59 

0.6009 

5985 

5961 

5938 

5914 

5890 

5867 

5844 

5820 

5797 

4 

8 

12 

16 

20 

60 

0.5774 

5750 

5727 

5704 

5681 

5658 

5635 

5612 

5589 

5566 

4 

8 

12 

15 

19 

61 

0.5543 

5520 

5498 

5475 

5452 

5430 

5407 

5384 

5362 

5340 

4 

8 

11 

15 

19 

62 

0.5317 

5295 

5272 

5250 

5228 

5206 

5184 

5161 

5139 

5117 

4 

7 

11 

15 

18 

63 

0.5095 

5073 

5051 

5029 

5008 

4986 

4964 

4942 

4921 

4899 

4 

7 

11 

15 

18 

64 

0.4877 

4856 

4834 

4813 

4791 

4770 

4748 

4727 

4706 

4684 

4 

7 

11 

14 

18 

65 

0.4663 

4642 

4621 

4599 

4578 

4557 

4536 

4515 

4494 

4473 

4 

7 

11 

14 

18 

66 

0.4452 

4431 

4411 

4390 

4369 

4348 

4327 

4307 

4286 

4265 

3 

7 

10 

14 

17 

67 

0.4245 

4224 

4204 

4183 

4163 

4142 

4122 

4101 

4081 

4061 

3 

7 

10 

14 

17 

68 

0.4040 

4020 

4000 

3979 

3959 

3939 

3919 

3899 

3879 

.3859 

3 

7 

10 

13 

17 

69 

0.3839 

3819 

3799 

3779 

3759 

3739 

3719 

3699 

.3679 

.3659 

3 

7 

10 

13 

17 

70 

0.3640 

.3620 

3600 

.3581 

.3561 

.3541 

3522 

.3502 

.3482 

.3463 

3 

7 

10 

13 

16 

71 

0.3443 

.3424 

3404 

.3385 

.3365 

.3346 

3327 

.3307 

.3288 

.3269 

3 

6 

10 

13 

16 

72 

0.3249 

.3230 

.3211 

.3191 

.3172 

.3153 

3134 

.3115 

.3096 

.3076 

3 

6 

10 

13 

16 

73 

0.3057 

.3038 

.3019 

.3000 

.2981 

.2962 

2943 

.2924 

.2905 

.2886 

3 

6 

9 

13 

16 

74 

0.2867 

.2849 

.2830 

.2811 

.2792 

.2773 

2754 

.2736 

.2717 

.2698 

3 

6 

9 

13 

16 

75 

0.2679 

.2661 

.2642 

.2623 

.2605 

.2586 

2568 

.2549 

.2530 

.2512 

3 

6 

12 

16 

76 

0.2493 

.2475 

.2456 

.2438 

.2419 

.2401 

2382 

.2364 

.2345 

.2327 

3 

6 

12 

15 

77 

0.2309 

.2290 

.2272 

.2254 

.2235 

.2217 

.2199 

.2180 

.2162 

.2144 

3 

6 

12 

15 

78 

0.2126 

.2107 

.2089 

.2071 

.2053 

.2035 

.2016 

.1998 

.1980 

.1962 

3 

6 

12 

15 

79 

0.1944 

.1926 

.1908 

.1890 

.1871 

.1853 

.1835 

.1817 

.1799 

.1781 

3 

6 

< 

12 

15 

80 

0.1763 

.1745 

.1727 

.1709 

.1691 

.1673 

.1655 

.1638 

.1620 

.1602 

3 

6 

12 

15 

81 

0.1584 

.1566 

.1548 

.1530 

.1512 

.1495 

.1477 

.1459 

.1441 

.1423 

3 

6 

12 

15 

82 

0.1405 

.1388 

.1370 

.1352 

.1334 

.1317 

.1299 

.1281 

.1263 

.1246 

3 

6 

12 

15 

83 

0.1228 

.1210 

.1192 

.1175 

.1157 

.1139 

.1122 

.1104 

.1086 

.1069 

3 

6 

12 

15 

84 

0.1051 

.1033 

.1016 

.0998 

.0981 

.0963 

.0945 

.0928 

.0910 

.0392 

3 

6 

12 

15 

85 

0.0875 

.0857 

.0840 

.0822 

.0805 

.0787 

.0769 

.0752 

.0734 

.0717 

3 

6 

12 

15 

86 

0.0699 

.0682 

.0664 

.0647 

.0629 

.0612 

.0594 

.0577 

.0559 

.0542 

3 

6 

12 

15 

87 

0.0524 

.0507 

.0489 

.0472 

.0454 

.0437 

.0419 

.0402 

.0384 

.0367 

3 

6 

12 

15 

88 

0.0349 

.0332 

.0314 

.0297 

.0279 

.0262 

.0244 

.0227 

.0209 

.0192 

3 

6 

12 

15 

89 

0.0175 

.0157 

.0140 

.0122 

.0105 

.0087 

.0070 

.0052 

.0035 

.0017 

3 

6 

12 

15 

All  cotangents  greater  than  cot  45°  are  less  than  1. 


150 


TABLES  FOE  REFERENCE 


POWERS  AND  ROOTS 


No. 

Squares 

Cubes 

Square 
Roots 

Cube 
Roots 

No. 

Squares 

Cubes 

Square 
Roots 

Cube 
Roots 

1 

1 

1 

1.000 

1.000 

51 

2601 

132  651 

7.141 

3.708 

2 

4 

8 

1.414 

1.260 

52 

2704 

140  608 

7.211 

3.733 

3 

9 

27 

1.732 

1.442 

53 

2809 

148  877 

7.280 

3.756 

4 

16 

64 

2.000 

1.587 

54 

2916 

157  464 

7.348 

3.780 

5 

25 

125 

2.236 

1.710 

55 

3025 

166  375 

7.416 

3.803 

6 

36 

216 

2.449 

1.817 

56 

3136 

175  616 

7.483 

3.826 

7 

49 

343 

2.646 

1.913 

57 

3249 

185  193 

7.550 

3.849 

8 

64 

512 

2.828 

2.000 

58 

3364 

195  112 

7.616 

3.871 

9 

81 

729 

3.000 

2.080 

59 

3481 

205  379 

7.681 

3.893 

10 

100 

1000 

3.162 

2.154 

60 

3600 

216  000 

7.746 

3.915 

11 

121 

1331 

3.317 

2.224 

61 

3721 

226  981 

7.810 

3.936 

12 

144 

1728 

3.464 

2.289 

62 

3844 

238  328 

7.874 

3.958 

13 

169 

2197 

3.606 

2.351 

63 

3969 

250  047 

7.937 

3.979 

14 

196 

2744 

3.742 

2.410 

64 

4096 

262  144 

8.000 

4.000 

15 

225 

3375 

3.873 

2.466 

65 

4225 

274  625 

8.062 

4.021 

16 

256 

4096 

4.000 

2.520 

66 

4356 

287  496 

8.124 

4.041 

17 

289 

4913 

4.123 

2.571 

67 

4489 

300763 

8.185 

4.062 

18 

324 

5832 

4.243 

2.621 

68 

4624 

314  432 

8.246 

4.082 

19 

361 

6859 

4.359 

2.668 

69 

4761 

328  509 

8.307 

4.102 

20 

400 

8000 

4.472 

2.714 

70 

4900 

343  000 

8.367 

4.121 

21 

441 

9261 

4.583 

2.759 

71 

5041 

357  911 

8.426 

4.141 

22 

484 

10648 

4.690 

2.802 

72 

5184 

373  248 

8.485 

4.160 

23 

529 

12167 

4.796 

2.844 

73 

5329 

389  017 

8.544 

4.179 

24 

576 

13824 

4.899 

2.884 

74 

5476 

405  224 

8.602 

4.198 

25 

625 

15625 

5.000 

2.924 

75 

5625 

421  875 

8.660 

4.217 

26 

676 

17576 

5.099 

2.962 

76 

5776 

438  976 

8.718 

4.236 

27 

729 

19683 

5.196 

3.000 

77 

5929 

456  533 

8.775 

4.254 

28 

784 

21952 

5.292 

3.037 

78 

6084 

474  552 

8.832 

4.273 

29 

841 

24389 

5.385 

3.072 

79 

6241 

493  039 

8.888 

4.291 

30 

900 

27000 

5.477 

3.107 

80 

6400 

512  000 

8.944 

4.309 

31 

961 

29791 

5.568 

3.141 

81 

6561 

531441 

9.000 

4.327 

32 

1024 

32768 

5.657 

3.175 

82 

6724 

551  368 

9.055 

4.344 

33 

1089 

35937 

5.745 

3.208 

83 

6889 

571  787 

9.110 

4.362 

34 

1156 

39304 

5.831 

3.240 

84 

7056 

592  704 

9.165 

4.380 

35 

1225 

42875 

5.916 

3.271 

85 

7225 

614  125 

9.220 

4.397 

36 

1296 

46656 

6.000 

3.302 

86 

7396 

636  056 

9.274 

4.414 

37 

1369 

50653 

6.083 

3.332 

87 

7569 

658  503 

9.327 

4.431 

38 

1444 

54872 

6.164 

3.362 

88 

7744 

681  472 

9.381 

4.448 

39 

1521 

59319 

6.245 

3.391 

89 

7921 

704  969 

9.434 

4.465 

40 

1600 

64000 

6.325 

3.420 

90 

8100 

729000 

9.487 

4.481 

41 

1681 

68921 

6.403 

3.448 

91 

8281 

753  571 

9.539 

4.498 

42 

1764 

74088 

6.481 

3.476 

92 

8464 

778  688 

9.592 

4.514 

43 

1849 

79507 

6.557 

3.503 

93 

8649 

804  357 

9.644 

4.531 

44 

1936 

85184 

6.633 

3.530 

94 

8836 

830  584 

9.695 

4.547 

45 

2025 

91125 

6.708 

3.557 

95 

9025 

857  375 

9.747 

4.563 

46 

2116 

97336 

6.782 

3.583 

96 

9216 

884  736 

9.798 

4.579 

47 

2209 

103  823 

6.856 

3.609 

97 

9409 

912  673 

9.849 

4.595 

48 

2304 

110  592 

6.928 

3.634 

98 

9604 

941  192 

9.899 

4.610 

49 

2401 

117649 

7.000 

3.659 

99 

9801 

970  299 

9.950 

4.626 

60 

2500 

125000 

7.071 

3.684 

100 

10000 

1  000  000 

10.000 

4.642 

TABLES  FOR  REFERENCE 


151 


SIZES  OF  TWIST  DRILLS  WITH  DECIMAL  EQUIVALENTS 


SIZE 

DECIMAL 
EQUIVALENT 

SIZE 

DECIMAL 
EQUIVALENT 

SIZE 

DECIMAL 
EQUIVALENT 

SIZE 

DECIMAL 
EQUIVALENT 

I" 

0.5000" 

4 

0.2500" 

#26 

0.1470" 

#56 

0.0465" 

ir 

.4844 

E 

.2500 

#27 

.1440 

#57 

.0430 

ji- 

.4688 

D 

.2460 

ft" 

.1406 

#58 

.0420 

ir 

.4531 

c 

.2420 

#28 

.1405 

#59 

.0410 

i7??" 

.4375 

B 

.2380 

#29 

.1360 

#60 

.0400 

ii" 

.4219 

si" 

.2344 

#30 

.1285 

#60^ 

.0390 

z 

.4130 

A 

.2340 

r 

.1250 

#61 

.0380 

ii" 

.4063 

#1 

.2280 

#31 

.1200 

#62 

.0370 

Y 

.4040 

#2 

.2210 

#32 

.1160 

#63 

.0360 

X 

.3970 

ft" 

.2188 

#33 

.1130 

#64 

.0350 

IT 

6  4: 

.3906 

#3 

.2130 

#34 

.1110 

#65 

.0330 

w 

.3860 

#4 

.2090 

#35 

.1100 

#66 

.0320 

V 

.3770 

#5 

.2055 

ft" 

.1094 

ft" 

.0313 

r 

.3750 

#6 

.2040 

#36 

.1065 

#67 

.0310 

U 

.3680 

ir 

.2031 

#37 

.1040 

#68 

.0300 

§|" 

.3594 

#7 

.2010 

#38 

.1015 

#681 

.0295 

T 

.3580 

#8 

.1990 

#39 

.0995 

#69 

.0290 

S 

.3480 

#9 

.1960 

#40 

.0980 

#69£ 

.0280 

jr 

.3438 

#10 

.1935 

#41 

.0960 

#70 

.0270 

R 

.3390 

#11 

.1910 

ft" 

.0938 

#71 

.0260 

Q 

.3320 

#12 

.1890 

#42 

.0935 

#71£ 

.0250 

ir 

.3281 

ft" 

.1875 

#43 

.0890 

#72 

.0240 

p 

.3230 

#13 

.1850 

#44 

.0860 

#73 

.0230 

0 

.3160 

#14 

.1820 

#45 

.•0820 

#73^ 

.0225 

ft" 

.3125 

#15 

.1800 

#46 

.0810 

#74 

.0220 

N 

.3020 

#16 

.1770 

#47 

.0785 

#74^ 

.0210 

if" 

.2969 

#17 

.1730 

ft" 

.0781 

#75 

.0200 

M 

.2950 

H" 

.1719 

#48 

.0760 

#76 

.0180 

L 

.2900 

#18 

.1695 

#49 

.0730 

#77 

.0160 

ft" 

.2813 

#19 

.1660 

#50 

.0700 

sV' 

.0156 

K 

.2810 

#20 

.1610 

#51 

.0670 

#78 

.0150 

J 

.2770 

#21 

.1590 

#52 

.0635 

#78^ 

.0145 

I 

.2720 

#22 

.1570 

A" 

.0625 

#79" 

.0140 

H 

.2660 

.5  " 

.1563 

#53 

.0595 

#79| 

.0135 

ir 

.2656 

#23 

.1540 

#54 

.0550 

#80 

.0130 

G 

.2610 

#24 

.1520 

#55 

.0520 

.  . 

.  .  . 

F 

.2570 

#25 

.1495 

ft" 

.0469 

•  • 

.  .  . 

152 


TABLES  FOR  REFERENCE 


SPEEDS  OF  DRILLS 


DIAMETER 
OF  DRILL 

WROUGHT 
IRON  AND 
STEEL 

CAST 
IRON 

BRASS 

DIAMETER 
OF  DRILL 

WROUGHT 
IRON  AND 
STEEL 

CAST 
IRON 

BRASS 

ft" 

1712 

2383 

3544 

1ft" 

72 

108 

180 

1 

855 

1191 

1772 

U 

68 

102 

170 

ft 

571 

794 

1181 

1ft 

64 

97 

161 

397 

565 

855 

U 

58 

89 

150 

T5ff 

318 

452 

684 

1ft 

55 

84 

143 

1 

265 

377 

570 

If 

53 

81 

136 

TV 

227 

323 

489 

1ft 

50 

77 

130 

4 

183 

267 

412 

14 

46 

74 

122 

T°ff 

163 

238 

367 

lT9ff 

44 

71 

117 

i 

147 

214 

330 

if 

40 

66 

113 

133 

194 

300 

»« 

38 

63 

109 

1 

112 

168 

265 

13 

37 

61 

105 

it 

103 

155 

244 

1T? 

36 

59 

101 

.£ 

96 

144 

227 

If 

33 

55 

98 

if 

89 

134 

212 

Hf 

32 

53 

95 

1 

76 

115 

191 

2 

31 

51 

92 

The  speeds  in  the  above  table  are  given  in  R.  P.  M. 


MACHINE  SCREW  THREADS,  OLD  STANDARD  SIZES 


NUMBER  OF 
SCREW 

OUTSIDE 
DIAMETER 

THREADS 

TO  1" 

NUMBER  OF 
SCREW 

OUTSIDE 
DIAMETER 

THREADS 

TO  1" 

1 

0.071" 

64 

12 

0.221" 

24 

14 

2 

.081 
.089 

56 
56 

13 
14 

.234 
.246 

22 
20 

3 

.101 

48 

15 

.261 

20 

4 

.113 

36 

16 

.272 

18 

5 

.125 

36 

18 

.298 

18 

6 

.141 

32 

20 

.325 

16 

7 

.154 

32 

22 

.350 

16 

8 

.166 

32 

24 

.378 

16 

9 

.180 

30 

26 

.404 

16 

10 

.194 

24 

28 

.430 

14 

11 

.206 

24 

30 

.456 

14 

The  basic  form  of  this  thread  is  the  same  as  that  of  a  U.S.S.  thread. 


TABLES  FOR  REFERENCE 


153 


MACHINE  SCREW  THREADS,  A.S.M.  E.  STANDARD  SIZES 


NUMBEK  OF 

SCREW 

MAXIMUM 
DIAMETER 

THREADS 

TOl" 

NUMBER  OF 
SCREW 

MAXIMUM 
DIAMETER 

THREADS 

TO  1" 

0 

0.060" 

80 

12 

0.216" 

28 

1 

.073 

72 

14 

.242 

24 

2 

.086 

64 

16 

.268 

22 

3 

.099 

56 

18 

.294 

20 

4 

.112 

48 

20 

.320 

20 

5 

.125 

44 

22 

.346 

18 

6 

.138 

40 

24 

.372 

16 

7 

.151 

36 

26 

.398 

16 

8 

.164 

36 

28 

.424 

14 

9 

.177 

32 

30 

.450 

14 

10 

.190 

30 

•    • 

-    • 

•    • 

MACHINE  SCREW  THREADS,  A.S.M.E.  SPECIAL  SIZES 


NUMBER  OF 
SCREW 

MAXIMUM 
DIAMETER 

THREADS 

TOl" 

NUMBER  OF 
SCREW 

MAXIMUM 
DIAMETER 

THREADS 

TO  1" 

1 

0.073" 

64 

9 

0.177" 

24 

2 

.086 

56 

10 

.190 

32 

3 

.099 

48 

10 

.190 

24 

4 

.112 

40 

12 

.216 

24 

4 

.112 

36 

14 

.242 

20 

5 

.125 

40 

16 

.268 

20 

5 

.125 

36 

18 

.294 

18 

6 

.138 

36 

20 

.320 

18 

6 

.138 

32 

22 

.346 

16 

7 

.151 

32 

24 

.372 

18 

7 

.151 

30 

26 

.398 

14 

8 

.164 

32 

28 

.424 

16 

8 

.164 

30 

30 

.450 

16 

9 

.177 

30 

•    • 

•    • 

The  A.S.M.E.  (American  Society  of  Mechanical  Engineers)  standard 
specifies  a  maximum  and  a  minimum  outside  diameter.  The  minimum 
diameter  (not  given  in  these  tables)  may  be  found  by  the  formula 

0.336 


Minimum  Dn  —  maximum  Dn  — 


40  +  number  of  threads  to  V 


154 


TABLES  FOR  REFERENCE 


U.S.S.-THREAD  BOLTS  AND  NUTS 


M 

. 

AREA  IN 

DIMENSIONS  OF  NUTS  AND 

H 

^ 

M 

^ 

SQUARE 

BOLT  HEADS 

H 

*"* 

^ 

^ 

INCHES 

M 

o 

H 

h 

5 

33 

a 

3 

1 

5 

O  k-3 

fl 

o 

O 

H 

H 

OUTBID 

8 

a 

J 

1 

1 

5 

P 

1 

| 

ft 

1 

j" 

20 

0.185'' 

w 

0.049 

0.026 

r 

0.578/x 

0.707" 

4~ 

4 

T6?r 

18 

0.240 

I 

0.076 

0.045 

0.686 

0.840 

A 

H 

| 

16 

0.294 

A 

0.110 

0.068 

11 

0.794 

0.972 

f 

T7s 

14 

0.345 

H 

0.150 

0.093 

it 

0.902 

1.105 

A 

I! 

| 

13 

0.400 

H 

0.196 

0.126 

1.011 

1.237 

i 

7 
T(T 

^ 

12 

0.454 

*f 

0.248 

0.162 

§£ 

1.119 

1.370 

* 

ii 

1 

11 

0.507 

0.307 

0.202 

*A 

1.227 

1.502 

17. 

10 

0.620 

fi 

0.442 

0.302 

4 

1.444 

1.768 

^ 

|" 

| 

9 

0.731 

0.601 

0.419 

1.660 

2.033 

-7 

1  •' 

1 

8 

0.838 

|5 

0.785 

0.551 

15 

1.877 

2.298 

1 

T§ 

H 

7 

0.939 

§^ 

0.994 

0.694 

HI 

2.093 

2.563 

H 

SI 

i^ 

7 

1.064 

1-^. 

1.227 

0.893 

2 

2.310 

2.828 

11 

if 

6 

1.158 

!A 

1.485 

1.057 

2  3 

2.527 

3.093 

]3 

1  *\ 

4 

6 

1.283 

1.767 

1.295 

23 
s 

2.743 

3.358 

H 

iA 

if 

ty 

1.389 

Iff 

2.074 

1.515 

2A 

2.960 

3.623 

If 

if 

5 

1.490 

1^1 

2.405 

1.746 

3.176 

3.889 

4 

i| 

i& 

5 

1.615 

1§^ 

2.761 

2.051 

2!-! 

3.393 

4.154 

4 

1  ),  f; 

2 

4^ 

1.711 

!fl 

3.142 

2.302 

3.609 

4.419 

2 

•'fV 

2i 

41 

1.961 

3.976 

3.023 

3£ 

4.043 

4.949 

2? 

4 

2* 

4 

2.175 

2^1 

4.909 

3.719 

3| 

4.476 

5.479 

2  i 

2| 

4 

2.425 

2fi 

5.940 

4.620 

4| 

4.909 

6.010 

2| 

21 

3 

3i 

2.629 

2T£ 

7.069 

5.428 

4 

5.342 

6.540 

3 

2f?r 

3^ 

4 

2.879 

2T| 

8,296 

6.510 

5 

5.775 

7.070 

3J 

2i 

3i 

3i 

3.100 

3fi 

9.621 

7.548 

51 

6.208 

7.600 

2H 

3f 

3 

3.317 

11.045 

8.641 

6.641 

8.131 

32 

4 

3 

3.567 

3| 

12.566 

9.963 

6^ 

7.074 

8.661 

4 

3iV 

4| 

2| 

3.798 

3fi 

14.186 

11.340 

6^ 

7.508 

9.191 

44 

3^    < 

4£ 

2| 

4.028 

4TJ\ 

15.904 

12.750 

6& 

7.941 

9.721 

4i 

3^ 

4| 

2I 

4.255 

4TV 

17.721 

14.215 

7i 

8.374* 

10.252 

4¥ 

s| 

5 

2| 

4.480 

4T9^ 

19.635 

15.760 

7f 

8.807 

10.782 

5 

3yii 

5^ 

2^ 

4.730 

413 

21.648 

17.570 

8 

9.240 

11.312 

5| 

4 

5i 

2i 

4.953 

^A 

23.758 

19.260 

8f 

9.673 

11.842 

51 

4fc 

5| 

2i 

5.203 

^  A 

25.967 

21.250 

8¥ 

10.106 

12.373 

5| 

4;^  ' 

6 

2i 

5.423 

5  2 

28.274 

23.090 

9I 

10.539 

12.903 

6 

4A 

TABLES  FOR  REFERENCE 


155 


TAP  DRILLS  FOR  MACHINE-SCREW  TAPS 


NUMBER 
OF  TAP 

THREADS 

TO  1" 

NUMBER  OF 
DRILL 

NUMBER 
OF  TAP 

THREADS 

TO  \" 

NUMBER  OF 
DRILL 

2 

48 

48 

13 

20 

17 

2 

56 

46 

13 

24 

15 

2 

64 

45 

14 

20 

14 

3 

40 

48 

14 

22 

13 

3 

48 

47 

14 

24 

11 

3 

56 

45 

15 

18 

12 

4 

32 

45 

15 

20 

10 

4 

36 

43 

15 

24 

7 

4 

40 

42 

5 
5 

30 
32 

41 
40 

16 
16 
16 

16 
18 
20 

10 
7 
5 

5 

36 

38 

16 

24 

1 

5 

40 

36 

6 
6 
6 

30 
32 
36 

39 
37 
35 

17 
17 
17 

16 
18 
20 

7 
4 
2 

6 

40 

33 

18 

16 

2 

7 

28 

32 

18 

18 

1 

7 

30 

31 

18 

20 

B 

7 

32 

30 

19 

16 

C 

8 

24 

31 

19 

18 

D 

8 

30 

30 

19 

20 

E 

8 

32 

29 

20 

16 

E 

9 

24 

29 

20 

18 

E 

9 

28 

27 

20 

20 

F 

9 

30 

26 

22 

16 

H 

9 

32 

24 

22 

18 

I 

10 

24 

26 

24 

14 

K 

10 

28 

24 

24 

16 

L 

10 

30 

23 

24 

18 

M 

10 

32 

21 

26 

14 

O 

11 

24 

20 

26 

16 

P 

11 

28 

19 

11 

30 

18 

28 

14 

R 

12 

20 

21 

28 

16 

S 

12 

22 

19 

30 

14 

T 

12 

24 

19 

30 

16 

U 

156 


TABLES  FOR  REFERENCE 


TAP  DRILLS  FOR  U.S.S.  THREADS 


THREAD 

THREADS 

DIAMETER 

THREAD 

THREADS 

DIAMETER 

DIAMETER 

TOl" 

OF  DRILL 

DIAMETER 

TOl" 

OF  DRILL 

¥ 

20 

0.191" 

1" 

8 

0.854" 

A 

18 

.248 

H 

7 

0.957 

i 

16 

.302 

u 

7 

1.082 

TV 

14 

.354 

If 

6 

1.179 

1 

13 

.409 

ll 

6 

1.304 

A 

12 

.465 

If 

H 

1.412 

i 

11 

.518 

if 

5 

1.515 

1 

10 

.632 

17 

5 

1.640 

7 

9 

.745 

2 

H 

1.739 

TAP  DRILLS  FOR  SHARP  V-THREADS 


THREAD 
DIAMETER 

THREADS 

TOl" 

DIAMETER 
OF  DRILL 

THREAD 
DIAMETER 

THREADS 

TOl" 

DIAMETER 
OF  DRILL 

i  f/ 

20 

0.184" 

1 

8 

0.832" 

A 

18 

.239 

1| 

7 

0.932 

1 

16 

.293 

I* 

7 

1.057 

7 
TS 

14 

.345 

1| 

6 

1.144 

i 

12 

.399 

H 

6 

1.269 

I9* 

.  12 

.453 

if 

5 

1.347 

f 

11 

.506 

if 

5 

1.472 

1 

10 

.618 

Ml 

^ 

1.566 

i 

9 

.728 

2 

Ji 

1.691 

TAP  DRILLS  FOR  BRIGGS  PIPE  THREADS 


NOMINAL 
SIZE 

THREADS 

TOl" 

DIAMETER 
OF  DRILL 

NOMINAL 
SIZE 

THREADS 

TOl" 

DIAMETER 
OF  DRILL 

I" 

8 

27 

0.328" 

1$" 

HJ 

1.719" 

£ 

18 

.453 

2 

111 

2.188 

I 

18 

.594 

2i 

8 

2.688 

14 

.719 

3 

8 

3.313 

a 

4 

14 

.938 

H 

8 

3.813 

1 

11J 

1.188 

4 

8 

4.313 

H 

Vj, 

1.469 

H 

8 

4.813 

The  diameter  of  drill  allows  for  reaming  the  hole  before  tapping. 


TABLES  FOR  REFERENCE 


157 


TOOTH  PARTS  OF  DIAMETRAL-PITCH  GEARS 


DIAMETRAL 
PITCH 

CIRCULAR 
PITCH 

THICKNESS 

OF  T(X)TH 

ON  PITCH 
LINE 

ADDENDUM 

WORKING 
DEPTH  OF 
TOOTH 

ROOT  OF 
TOOTH 

WHOLE 
DEPTH  OF 
TOOTH 

1 

6.2832" 

3.1416" 

2.0000" 

4.0000" 

2.3142" 

4.3142" 

a 

4 

4.1888 

2.0944 

1.3333 

2.6666 

1.5428 

2.8761 

1 

3.1416 

1.5708 

1.0000 

2.0000 

1.1571 

2.1571 

1| 

2.5133 

1.2566 

0.8000 

1.6000 

0.9257 

1.7257 

2.0944 

1.0472 

.6666 

1.3333 

.7714 

1.4381 

1| 

1.7952 

0.8976 

.5714 

1.1429 

.6612 

1.2326 

2 

1.5708 

.7854 

.5000 

1.0000 

.5785 

1.0785 

21 

1.3963 

.6981 

.4444 

0.8888 

.5143 

0.9587 

25 

1.2566 

.6283 

.4000 

.8000 

.4628 

.8628 

2| 

1.1424 

.5712 

.3636 

.7273 

.4208 

.7844 

3 

1.0472 

.5236 

.3333 

.6666 

.3857 

.7190 

3S 

0.8976 

.4488 

.2857 

.5714 

.3306 

.6163 

4 

.7854 

.3927 

.2500 

.5000 

.2893 

.5393 

5 

.6283 

.3142 

.2000 

.4000 

.2314 

.4314 

6 

.5236 

.2618 

.1666 

.3333 

.1928 

.3595 

7 

.4488 

.2244 

.1429 

.2857 

.1653 

.3081 

8 

.3927 

.1963 

.1250 

.2500 

.1446 

.2696 

9 

.3491 

.1745 

.1111 

.2222 

.1286 

.2397 

10 

.3142 

.1571 

.1000 

.2000 

.1157 

.2157 

12 

.2618 

.1309 

.0833 

.1666 

.0964 

.1798 

14 

.2244 

.1122 

.0714 

.1429 

.0826 

.1541 

16 

.1963 

.0982 

.0625 

.1250 

.0723 

.1348 

18 

.1745 

.0873 

.0555 

.1111 

.0643 

.1198 

20 

.1571 

.0785 

.0500 

.1000 

.0579 

.1079 

22 

.1428 

.0714 

.0455 

.0909 

.0626 

.0980 

24 

.1309 

.0654 

.0417 

.0833 

.0482 

.0898 

26 

.1208 

.0604 

.0385 

.0769 

.0445 

.0829 

28 

.1122 

.0561 

.0357 

.0714 

.0413 

.0770 

30 

.1047 

.0524 

.0333 

.0666 

.0386 

.0719 

32 

.0982 

.0491 

.0312 

.0625 

.0362 

.0674 

36 

.0873 

.0436 

.0278 

.0555 

.0321 

.0599 

40 

.0785 

.0393 

.0250 

.0500 

.0289 

.0539 

44 

.0714 

.0357 

.0227 

.0455 

.0263 

.0490 

48 

.0654 

.0327 

.0208 

.0417 

.0241 

.0449 

50 

.0628 

.0314 

.0200 

.0400 

.0231 

.0431 

158 


TABLES  FOR  REFERENCE 


TOOTH  PARTS  OF  CIRCULAR-PITCH  GEARS 


CIRCULAR 
PITCH 

DIAMETRAL 
PITCH 

THICKNESS 
OF  TOOTH 
ON  PITCH 
LINE 

ADDENDUM 

WORKING 
DEPTH  OF 

To<  )TH 

ROOT  OF 
TOOTH 

WHOLE 
DEPTH  OF 
TOOTH 

4" 

0.7854 

2.0000" 

1.2732" 

2.5464" 

1.4732" 

2.7464" 

3i 
3 

0.8976 
1.0472 

1.7500 
1.5000 

1.1140 
0.9549 

2.2281 
1.9098 

1.2890 
1.1049 

2.4031 
2.0598 

2  4 

1.1424 

1.3750 

.8753 

1.7506 

1.0128 

1.8881 

2, 

1.2566 
1.3963 

1.2500 
1.1250 

.7957 
.7162 

1.5915 
1.4323 

0.9207 

.8287 

1.7165 
1.5448 

2 

1.5708 

1.0000 

.6366 

1.2732 

.7366 

1.3732 

1J 

1.6755 

0.9375 

.5968 

1.1937 

.6906 

1.2874 

If 
If 

1.7952 
1.9333 

.8750 
.8125 

.5570 
.5173 

1.1141 
1.0345 

.6445 
.5985 

1.2016 
1.1158 

** 

2.0944 
2.1855 

.7500 
.7187 

.4775 
.4576 

0.9549 
.9151 

.5525 
.5294 

1.0299 
0.9870 

If* 

2.2848 

.6875 

.4377 

.8754 

.5064 

.9441 

2.3936 

.6562 

.4178 

.8356 

.4834 

.9012 

H 

2.5133 
2.6456 

.6250 
.5937 

.3979 
.3780 

.7958 
.7560 

.4604 
.4374 

.8583 
.8154 

11 

2.7925 

.5625 

.3581 

.7162 

.4143 

.7724 

^A 

2.9568 

.5312 

.3382 

.6764 

.3913 

.7295 

i 

3.1416 

.5000 

.3183 

.6366 

.3683 

.6866 

if 

7 

8 

H 

3 

4 
U 
f 

3.3510 
3.5904 
3.8666 
4.1888 
4.5696 
5.0265 

.4687 
.4375 
.4062 
.3750 
.3437 
.3125 

.2984 
.2785 
.2586 
.2387 
.2189 
.1989 

.5968 
.5570 
.5173 
.4775 
.4377 
.3979 

.3453 
.3223 
.2993 
.2762 
.2532 
.2301 

.6437 
.6007 
.5579 
.5150 
.4720 
.4291 

5.5851 

.2812 

.1790 

.3581 

.2071 

.3862 

1 

6.2832 

.2500 

.1592 

.3183 

.1842 

.3433 

TV 

7.1808 

.2187 

.1393 

.2785 

.1611 

.3003 

3 

8.3776 

.1875 

.1194 

.2387 

.1381 

.2575 

1. 

9.4248 

.1666 

.1061 

.2122 

.1228 

.2289 

T5ff 

10.0531 

.1562 

.0995 

.1989 

.1151 

.2146 

1 

12.5664 

.1250 

.0796 

.1591 

.0921 

.1716 

3 

16.7552 

.0937 

.0597 

.1194 

.0690 

.1287 

1 

25.1327 

.0625 

.0398 

.0796 

.0460 

.0858 

TV 

50.2655 

.0312 

.0199 

.0398 

.0230 

.0429 

INDEX 


PAGE 

Acme  thread 48 

Addendum 86 

angle 98 

angular 98 

Angle 28,  138 

addendum 98 

with  axis 28,  32,  33 

clearance 46 

cutting 98 

dedendum 98 

diameter 52 

face 98,  112 

gashing 112,  115 

included 28,  32,  33 

for  milling  cams  ....  82 

pitch-cone 99 

spiral 73,  78,  104 

tooth 104 

Angular  addendum 98 

indexing 66 

Axial  pitch 104 


Bevel  gear     .    .    . 
Briggs  pipe  thread 


.    .      97 
.  50,  156 


Caliper 

micrometer  .  . 
with  vernier 

vernier  .... 
Cam  . 


.  .  2 
.  .  2 
.  .  5 

4 

80 

Center  distance     .    .    .    86,  91,  103 
Change  gears   56,  57,  70,  73,  75,  107 


PAGE 

Chordal  pitch 86 

Circle,  outside 86 

pitch 85,  86,  98,  99 

Circular  pitch  ....  86,  97,  104 
Clearance  ....  46,  86,  113,  115 
Compound  gearing  .  .  .59,  72,  75 

indexing 67 

rest 25,  28 

Cutter,  gear 107 

Cutting  angle 98 

speeds     ...    9,  10,  11,  12,  16 

Decimal  equivalents     ....    139 

Dedendum 86 

angle 98 

Depth,  double 35 

of  thread 35 

whole 87,  90,  97 

working 87 

Diagonal  scale 1 

Diameter,  angle 52 

increment 98 

outside 35,  86,  97 

pitch  ....  52,  85,  86,  97,  99 

root 35 

throat     ........    112 

Diametral  pitch  ...  86,  97,  104 
Differential  indexing  ....  70 

Direct  indexing 62 

Dividing  head 61,  63,  73 

Drills,  sizes  of  . 151 

speeds  of 13,  152 


159 


160 


INDEX 


Equivalents,  common 
decimal  . 


PACK 

140 

139 


Face 86,  99 

angle 98,  112 

Feed 9,  17 

Finishing  speeds 10 

Flank 86 

Follower 80 

Fractions,  equivalents  of  ...    139 

indexing 69,70 

Franklin  Institute  thread     .    .      41 

French  thread 44 

Friction  surface    .    .    .    .85,  97,  99 
Functions,  tables  of  ...     141-149 

Gaging  depth  of  cut 8 

Gashing  angle 112,  115 

Gearing,  compound  .    .    .  59,  72,  75 

simple 56,  71 

worm 112 

Gears .      85 

bevel 97 

change    .    .  56,  57,  70,  73, 

75,  107 

cutters  for 107 

helical 103 

internal 85 

spiral 103,  110 

spur 85,  86 

teeth  of  ...     85,  94,  157,  158 

Hob 115 

Robbing 113,  115 

Increment,  diameter     ....  98 

Index  center 62 

head 61 

plates 65 

Indexing 61 

angular 66 


PAGE 
Indexing  (continued) 

compound 67 

differential 70 

direct 62 

fractions     .    .    .    .    .    .    69,  70 

simple 64 

Instruments,  measuring    ...        1 

International  thread     ....      44 

Involute  teeth   .    .    .    ;-- ;-— r  .      85 

sizes  of   .    .  ...    .    .94,  157,  158 

Lathe,  compound  gearing  for  .  59 
cutting  speed  of  ....  10 
simple  gearing  for  ...  56 

Lead  of  a  cam 80 

of  a  milling  machine  .  .  75 

of  a  screw 35 

of  a  spiral  ....  73,  75,  104 
screw 56 

Linear  pitch 96,  112 

Machine  screw  thread    152,  153,  155 

Measures,  common 137 

metric     .    .,  ,. 138 

Measuring  instruments     ...  1 

tapers .    .    32,  33 

threads 52 

Metric  measures 138 

thread 44 

Micrometer .  2 

caliper $/ 

with  vernier    ....  5 

reading  a 8 

Milling  machine,  cutting  speed 

of 12 

lead  of 75 

setting  table  of 78 

Multiple  thread 36 

Normal  circular  pitch  ....  104 
diametral  pitch 104 


INDEX 


161 


PAGE 

Offset  for  tapers 24 

Outside  circle 86 

diameter 35,  86,  97 

Pitch 35,  86,  103 

axial 104 

chordal 86 

circle 85,  86,  98,  99 

circular 86,  97,  104 

-cone  alible 99 

radius 99 

diameter  of  a  gear  85,  86,  97,  99 

of  a  thread 52 

diametral   ....    86,  97,  104 

line 86,  98,  99 

linear 96,  112 

normal 104 

Pinion 85,  91,  97 

Pipe  thread 50,  156 

Planer,  cutting  speed  of    ...      11 
Problem  material 

bevel  gears     ....     102,  116 

calipers 3,  6 

cams  ........    83,  84 

cutting  speeds     10,  11,  12, 

14,  16,  18,  20 

drill  press 118 

drilling  machine  ....  126 
drills,  speeds  of  .  .  13,  14,  20 

feeds 17,  18 

gaging  cuts 8 

general  applications  .117, 

134,  136 

indexing  .  65,  66,  69,  72,  120 
lathes  ....  10,  14,  20,  132 
milling  .  .  12,  14,  18,  120,  128 

planers 11,  18,  124 

review      14,  18,  30,  55,  72, 

84,  96,  117-136 

shapers 16,  20 

spiral  gears     .    .     109,  111,  116 


PAGE 

Problem  material  (continued) 

spirals     '. 77,  79,  84 

spur  gears  .  89,  92,  94,  96,  116 
tapers   .  .  22,  26,  29,  30, 

34,  122,  130 
threads  37,  39,  41,  43,  45, 

47,  49,  51,  54,  56,  58,  60 
worm  gearing     .    .    .     115,  116 

Rack 85,  96 

Radius,  pitch-cone 99 

Ratio,  velocity 91,  106 

Rest,  compound 25,  28 

Review  problems  .     14,  18,  30, 

55,  72,  84,  96,  117-136 

Rise 80 

Root 35,  86 

diameter 35 

Roughing  speeds 10 

Rules,  convenient 140 

S.  A.  E.  thread      40 

Scale,  diagonal 1 

Screw,  hand  of 36 

lead  of 35 

thread 35 

Sellers  thread 41 

Shaper,  cutting  speed  of  ...      16 

Sharp  V-thread 38 

Simple  gearing 56,  71 

indexing 64 

Sine  bar 33 

Speeds,  cutting      .    9,  10,  11,  12,  16 

of  drills 13,  152 

surface 9 

Spiral 73 

angle 73,  78,  104 

gears 103 

with  parallel  shafts     .    110 

head 61 

lead  of 73,  75,  104 


162 


INDEX 


PAGE 

Spur  gears 85,  86 

Square  thread 46 

tool 46 

Stud  ! 56,  59 

Symbols     .    .     21,  88,  100,  106,  112 

Tables 

angles  and  arcs 138 

Briggs  pipe  threads    .    .  51,  156 
common  fractions  ....    139 

measures 137 

cosines 144 

cotangents 148 

cutting  speeds    .    .    .    .    10,  12 
decimal  equivalents   .    .    .    139 

drills,  sizes  of 151 

speeds  of 152 

feeds 17 

machine  screw  threads  152, 

153,  155 
metric  measures     ....    138 

threads 44 

powers  and  roots    ....    150 

S.  A.  E.  thread 40 

sines 142 

tangents 146 

tap  drills    .....     155,  156 

tooth  parts     ....     157,  158 

U.  S.  S.  thread  ....  40,  156 

bolts  and  nuts     .    .    .    154 

V-thread 38,  156 

Whitworth  thread      ...      42 

Tailstock,  offsetting  the    ...      24 

Tap 39 

-drill  size    .    .     39,  41,  155,  156 

Taper 21 

attachment 25 

measuring  a 32,  33 

testing  a 32 

turning  a    ...'.-...      24 


PAGE 

Teeth,  cycloidal 85 

involute 85,  95 

sizes  of  ...  94,  157,  158 

Thread 35 

Acme j  .  48 

cutting 56,  59 

machine  screw  .  152,  153,  155 

measuring 52 

metric 44 

multiple  .  .. 36 

pipe  ........  50,  156 

S.  A.  E 40 

screw 35 

square 46 

U.  S.  S.  .  .  .  40,  52,  154,  156 

V- 38,  53,  156 

Whitworth 42,  54 

Throat  diameter 112 

Tooth  angle 104 

thickness  of  .  87,  100,  102,  107 

Trigonometric  functions  ...    141 


U.  S.  S.  thread 


40,  52,  154,  156 


V-thread,  sharp    ...    38,  53,  156 

Velocity  ratio 91,  106 

Vernier 4 

caliper 4 

micrometer  caliper  with    .        5 

reading  a 5 

Vertex  of  a  bevel  gear  ....      99 

Wheel,  gear 85 

worm  .......    63,  64,  112 

Whitworth  thread     ....    42,  54 

Whole  depth 87,  90,  97 

Working  depth 87 

Worm 63,  64,  73,  112 

gearing 112 

wheel 63,  64,  112 


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